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I'll give an answer, only because I'm interested in chasing down these references myself. But all I'm doing is assembling references. I assume that BCnrd will keep me honest. [July 21: I've added some remarks about constructibility, to makes this more useful (at least to me).]

Since I'm a complex geometer rather an arithmetic one, let me start with the first case for intuition. If $X_{an}$ is a (connected) complex variety endowed with the classical topology then one knows that representations of the usual $\pi_1(X_{an},x)$ correspond to locally constant sheaves on $X_{an}$. This is classical. A good source of examples are as follows: if $f:Y\to X$ is a surjective smooth proper map, then it is topologically a fibre bundle (Ereshmann). Therefore $R^if_*\mathbb{Z}$ is locally constant. The corresponding $\pi_1(X)$-module is the monodromy representation. The most general statement one can make, without making any assumptions on $f$, is that the proper direct image $R^if_!\mathbb{Z}$ is constructible. Note that constructibility can mean different things in the topological world. The best notion (from my point of view) is what is sometimes called algebraic constructibility: there exists a partition of the base into Zariski locally closed strata such that the restrictions of the sheaf are locally constant. The only reference that I know which takes this viewpoint is Verdier, Classe d'homologie associée à un cycle. If people are aware of other sources, please let me know.

Remarkably, the analogous results hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field. A lisse (resp. constructible) $\ell$-adic sheaf is now a prosheaf $$\ldots \mathcal{F}_n\to \mathcal{F}_{n-1}\ldots $$ on the etale sie $X_{et}$ such that each item above is a locally constant (resp. constructible) $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). For lisse sheaves, each $\mathcal{F}_n$ gives a representation of the etale fundamental group $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}/\ell^n)$$ ($x$ a geom. pt.). So passing to the limit, we get a continuous representation $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}_\ell)$$ This constuction is an equivalence [FK,p 286].

The corresponding result that $R^if_*\mathbb{Z}_\ell$ is lisse, when $f$ is smooth, proper and surjective, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website. The contrucibility of $R^if_!\mathbb{Z}_\ell$ would follow from SGA4 exp XIV 1.1 (It ought to be in [FK,M], but I probably didn't look hard enough.)

When $X$ is defined over $\mathbb{C}$, one can compare cohomology for the classical and etale topologies with general coefficients by applying SGA4 exp XVI 4.1 and taking inverse limits. A more general comparison result for the "6 operations" is given in [Beilinson-Bernstein-Deligne p 150], but the proof seems a bit sketchy.

I'll give an answer, only because I'm interested in chasing down these references myself. But all I'm doing is assembling references. I assume that BCnrd will keep me honest. [July 21: I've added some remarks about constructibility, to makes this more useful (at least to me).]

Since I'm a complex geometer rather an arithmetic one, let me start with the first case for intuition. If $X_{an}$ is a (connected) complex variety endowed with the classical topology then one knows that representations of the usual $\pi_1(X_{an},x)$ correspond to locally constant sheaves on $X_{an}$. This is classical. A good source of examples are as follows: if $f:Y\to X$ is a surjective smooth proper map, then it is topologically a fibre bundle (Ereshmann). Therefore $R^if_*\mathbb{Z}$ is locally constant. The corresponding $\pi_1(X)$-module is the monodromy representation. The most general statement one can make, without making any assumptions on $f$, is that the proper direct image $R^if_!\mathbb{Z}$ is constructible. Note that constructibility can mean different things in the topological world. The best notion (from my point of view) is what is sometimes called algebraic constructibility: there exists a partition of the base into Zariski locally closed strata such that the restrictions of the sheaf are locally constant. The only reference that I know which takes this viewpoint is Verdier, Classe d'homologie associée à un cycle. If people are aware of other sources, please let me know.

Remarkably, the analogous results hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field. A lisse (resp. constructible) $\ell$-adic sheaf is now a prosheaf $$\ldots \mathcal{F}_n\to \mathcal{F}_{n-1}\ldots $$ on the etale site $X_{et}$ such that each item above is a locally constant (resp. constructible) $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). For lisse sheaves, each $\mathcal{F}_n$ gives a representation of the etale fundamental group $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}/\ell^n)$$ ($x$ a geom. pt.). So passing to the limit, we get a continuous representation $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}_\ell)$$ This constuction is an equivalence [FK,p 286].

The corresponding result that $R^if_*\mathbb{Z}_\ell$ is lisse, when $f$ is smooth, proper and surjective, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website. The contrucibility of $R^if_!\mathbb{Z}_\ell$ would follow from SGA4 exp XIV 1.1 (It ought to be in [FK,M], but I probably didn't look hard enough.)

When $X$ is defined over $\mathbb{C}$, one can compare cohomology for the classical and etale topologies with general coefficients by applying SGA4 exp XVI 4.1 and taking inverse limits. A more general comparison result for the "6 operations" is given in [Beilinson-Bernstein-Deligne p 150], but the proof seems a bit sketchy. Remark added July 22: Unfortunately, this part of the story appears to be inadequately addressed in the literature. See BCnrd's comment below.

I'll give an answer, only because I'm interested in chasing down these references myself. But all I'm doing is assembling references. I assume that BCnrd will keep me honest. [July 21: I've added some remarks about constructibility, to makes this more useful (at least to me).]

Since I'm a complex geometer rather an arithmetic one, let me start with the first case for intuition. If $X_{an}$ is a (connected) complex variety endowed with the classical topology then one knows that representations of the usual $\pi_1(X_{an},x)$ correspond to locally constant sheaves on $X_{an}$. This is classical. A good source of examples are as follows: if $f:Y\to X$ is a surjective smooth proper map, then it is topologically a fibre bundle (Ereshmann). Therefore $R^if_*\mathbb{Z}$ is locally constant. The corresponding $\pi_1(X)$-module is the monodromy representation. The most general statement one can make, without making any assumptions on $f$, is that the proper direct image $R^if_!\mathbb{Z}$ is constructible. Note that constructibility can mean different things in the topological world. The best notion (from my point of view) is what is sometimes called algebraic constructibility: there exists a partition of the base into Zariski locally closed strata such that the restrictions of the sheaf are locally constant. The only reference that I know which takes this viewpoint is Verdier, Classe d'homologie associée à un cycle. If people are aware of other sources, please let me know.

Remarkably, the analogous results hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field. A lisse (resp. constructible) $\ell$-adic sheaf is now a prosheaf $$\ldots \mathcal{F}_n\to \mathcal{F}_{n-1}\ldots $$ on the etale sie $X_{et}$ such that each item above is a locally constant (resp. constructible) $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). Each $\mathcal{F}_n$ gives a representation of the etale fundamental group $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}/\ell^n)$$ ($x$ a geom. pt.). So passing to the limit, we get a continuous representation $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}_\ell)$$ This constuction is an equivalence [FK,p 286].

The corresponding result that $R^if_*\mathbb{Z}_\ell$ is lisse, when $f$ is smooth, proper and surjective, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website. The contrucibility of $R^if_!\mathbb{Z}_\ell$ would follow from SGA4 exp XIV 1.1 (It ought to be in [FK,M], but I probably didn't look hard enough.)

When $X$ is defined over $\mathbb{C}$, one can compare cohomology for the classical and etale topologies with general coefficients by applying SGA4 exp XVI 4.1 and taking inverse limits. A more general comparison result for the "6 operations" is given in [Beilinson-Bernstein-Deligne p 150], but the proof seems a bit sketchy.

I'll give an answer, only because I'm interested in chasing down these references myself. But all I'm doing is assembling references. I assume that BCnrd will keep me honest. [July 21: I've added some remarks about constructibility, to makes this more useful (at least to me).]

Since I'm a complex geometer rather an arithmetic one, let me start with the first case for intuition. If $X_{an}$ is a (connected) complex variety endowed with the classical topology then one knows that representations of the usual $\pi_1(X_{an},x)$ correspond to locally constant sheaves on $X_{an}$. This is classical. A good source of examples are as follows: if $f:Y\to X$ is a surjective smooth proper map, then it is topologically a fibre bundle (Ereshmann). Therefore $R^if_*\mathbb{Z}$ is locally constant. The corresponding $\pi_1(X)$-module is the monodromy representation. The most general statement one can make, without making any assumptions on $f$, is that the proper direct image $R^if_!\mathbb{Z}$ is constructible. Note that constructibility can mean different things in the topological world. The best notion (from my point of view) is what is sometimes called algebraic constructibility: there exists a partition of the base into Zariski locally closed strata such that the restrictions of the sheaf are locally constant. The only reference that I know which takes this viewpoint is Verdier, Classe d'homologie associée à un cycle. If people are aware of other sources, please let me know.

Remarkably, the analogous results hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field. A lisse (resp. constructible) $\ell$-adic sheaf is now a prosheaf $$\ldots \mathcal{F}_n\to \mathcal{F}_{n-1}\ldots $$ on the etale sie $X_{et}$ such that each item above is a locally constant (resp. constructible) $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). For lisse sheaves, each $\mathcal{F}_n$ gives a representation of the etale fundamental group $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}/\ell^n)$$ ($x$ a geom. pt.). So passing to the limit, we get a continuous representation $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}_\ell)$$ This constuction is an equivalence [FK,p 286].

The corresponding result that $R^if_*\mathbb{Z}_\ell$ is lisse, when $f$ is smooth, proper and surjective, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website. The contrucibility of $R^if_!\mathbb{Z}_\ell$ would follow from SGA4 exp XIV 1.1 (It ought to be in [FK,M], but I probably didn't look hard enough.)

When $X$ is defined over $\mathbb{C}$, one can compare cohomology for the classical and etale topologies with general coefficients by applying SGA4 exp XVI 4.1 and taking inverse limits. A more general comparison result for the "6 operations" is given in [Beilinson-Bernstein-Deligne p 150], but the proof seems a bit sketchy.

I'll give an answer, only because I'm interested in chasing down these references myself, but there's not much content. Anyone could do this. I assume that BCnrd will keep me honest.

Since I'm a complex geometer rather an arithmetic one, let me start with the first case for intuition. If $X_{an}$ is a (connected) complex variety endowed with the classical topology then one knows that representations of the usual $\pi_1(X_{an},x)$ correspond to locally constant sheaves on $X_{an}$. This is classical. A good source of examples are as follows: if $f:Y\to X$ is a surjective smooth proper map, then it is topologically a fibre bundle (Ereshmann). Therefore $R^if_*\mathbb{Z}$ is locally constant. The corresponding $\pi_1(X)$-module is the monodromy representation.

Remarkably, the analogous result hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field. A lisse $\ell$-adic sheaf is now a prosheaf $$\ldots \mathcal{F}_n\to \mathcal{F}_{n-1}\ldots $$ on the etale sie $X_{et}$ such that each item above is a locally constant $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). Each $\mathcal{F}_n$ gives a representation of the etale fundamental group $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}/\ell^n)$$ ($x$ a geom. pt.). So passing to the limit, we get a continuous representation $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}_\ell)$$ This constuction is an equivalence [FK,p 286].

The corresponding result that $R^if_*\mathbb{Z}_\ell$ is lisse, when $f$ is smooth, proper and surjective, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website.

When $X$ is defined over $\mathbb{C}$, one can compare cohomology for the classical and etale topologies with general coefficients by applying SGA4 exp XVI 4.1 and taking inverse limits. A more general comparison result for the "6 operations" is given in [Beilinson-Bernstein-Deligne p 150], but the proof seems a bit sketchy.

I'll give an answer, only because I'm interested in chasing down these references myself. But all I'm doing is assembling references. I assume that BCnrd will keep me honest. [July 21: I've added some remarks about constructibility, to makes this more useful (at least to me).]

Since I'm a complex geometer rather an arithmetic one, let me start with the first case for intuition. If $X_{an}$ is a (connected) complex variety endowed with the classical topology then one knows that representations of the usual $\pi_1(X_{an},x)$ correspond to locally constant sheaves on $X_{an}$. This is classical. A good source of examples are as follows: if $f:Y\to X$ is a surjective smooth proper map, then it is topologically a fibre bundle (Ereshmann). Therefore $R^if_*\mathbb{Z}$ is locally constant. The corresponding $\pi_1(X)$-module is the monodromy representation. The most general statement one can make, without making any assumptions on $f$, is that the proper direct image $R^if_!\mathbb{Z}$ is constructible. Note that constructibility can mean different things in the topological world. The best notion (from my point of view) is what is sometimes called algebraic constructibility: there exists a partition of the base into Zariski locally closed strata such that the restrictions of the sheaf are locally constant. The only reference that I know which takes this viewpoint is Verdier, Classe d'homologie associée à un cycle. If people are aware of other sources, please let me know.

Remarkably, the analogous results hold in the $\ell$-adic case, although for different reasons. Let $X$ be variety over some field. A lisse (resp. constructible) $\ell$-adic sheaf is now a prosheaf $$\ldots \mathcal{F}_n\to \mathcal{F}_{n-1}\ldots $$ on the etale sie $X_{et}$ such that each item above is a locally constant (resp. constructible) $\mathbb{Z}/\ell^n$-module etc. (see Freitag-Kiehl, pp 118-131, for the precise conditions). Each $\mathcal{F}_n$ gives a representation of the etale fundamental group $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}/\ell^n)$$ ($x$ a geom. pt.). So passing to the limit, we get a continuous representation $$\pi_1^{et}(X,x)\to GL_N(\mathbb{Z}_\ell)$$ This constuction is an equivalence [FK,p 286].

The corresponding result that $R^if_*\mathbb{Z}_\ell$ is lisse, when $f$ is smooth, proper and surjective, should follow from Theorem 20.2 of Milne "Lectures on etale cohomology" from his website. The contrucibility of $R^if_!\mathbb{Z}_\ell$ would follow from SGA4 exp XIV 1.1 (It ought to be in [FK,M], but I probably didn't look hard enough.)

When $X$ is defined over $\mathbb{C}$, one can compare cohomology for the classical and etale topologies with general coefficients by applying SGA4 exp XVI 4.1 and taking inverse limits. A more general comparison result for the "6 operations" is given in [Beilinson-Bernstein-Deligne p 150], but the proof seems a bit sketchy.