Topology of function spaces? Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected.  In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,Y)$ denote the space of smooth maps $f: X \to Y$.  I'm interested in, say, the connected components, fundamental group,... of this space, but I'm really not sure where to start looking.  I realise that I first need to topologise this space.  My only experience in this realm is an introductory point-set topology course (based on Munkres) I took as a graduate student.  Munkres talks about the compact-open topology for the space $C^0(X,Y)$ of continuous maps between two topological spaces and shows, for instance, that if $X$ is locally compact Hausdorff then the evaluation map $X \times C^0(X,Y) \to Y$ is continuous.  Later in the book he also applies this to give a slick proof of the existence of covering spaces with prescribed covering group.
Back to the differentiable category, ideally I'd like to be able to do calculus on $C^\infty(X,Y)$, hence I'd like to think of $C^\infty(X,Y)$ as an infinite-dimensional differentiable manifold and possibly even riemannian whenever so are $X$ and $Y$.
In case it helps to focus the question, let me say a few words of (physical, I fear) motivation.
When $X,Y$ are riemannian, $C^\infty(X,Y)$ plays the rôle of the configuration space for a physical model known as the nonlinear sigma model, whose action functional, assigning to $\sigma: X \to Y$, the value of the integral (either take $X$ to be compact or else restrict the possible functions further to assure convergence)
$$ S[\sigma] = \int_X |d\sigma|^2 \operatorname{dvol}_X,$$
where I'd like to think of $d\sigma$ as a one-form on $X$ with values in the pullback $\sigma^*TY$ by $\sigma$ of the tangent bundle to $Y$, and $|d\sigma|^2$ involves the metric on the bundle $T^*X \otimes \sigma^*TY$ induced from the riemannian metrics on $X$ and on $Y$.  The extrema of $S$ are then the harmonic maps.
We are often interested in the quantum theory (un)defined formally by a path integral.  A mathematically conservative point of view is that the path integral simply gives a recipe for the perturbative treatment of the quantum theory, where we fix an extremum $\sigma_0$ of $S$ and quantise the fluctuations around $\sigma_0$.  By definition, fluctuations around $\sigma_0$ lie in the connected component of $\sigma_0$ and as a first approximation, the path integral becomes a sum over the connected components of the space of maps.  Hence the interest in determining the connected components of $C^\infty(X,Y)$, which in this context are often called superselection sectors.
So in summary, a possible question would be this:

What can be said about the topology (e.g., homotopy type) of $C^\infty(X,Y)$ in terms of $X$ and $Y$?

I'm not asking for a tutorial, just for some orientation to the available literature.
Thanks in advance.
Added
In response to the helpful answers I have received already, I'd like to point out that my interest is really on the homotopy type of the space of maps.  I only mentioned the analytic aspects in case that narrows down the topology one would put on the space.  As pointed out in the answers, most reasonable topologies are equivalent, so this is a relief.
As for concrete examples, I am particularly interested in the case where $X$ is a compact Riemann surface and $Y$ a compact Lie group.  I'm happy to put the constant curvature metric on $X$ and a bi-invariant metric on $Y$.
 A: The rational homotopy type of $C^\infty(X,Y)$ or $C^0(X,Y)$ has been determined by 


*

*MR0667163 (84a:55010) 
Haefliger, André:
Rational homotopy of the space of sections of a nilpotent bundle. 
Trans. Amer. Math. Soc. 273 (1982), no. 2, 609–620. 


See also


*

*MR0850369 (87h:55009)
Vigué-Poirrier, Micheline:
Sur l'homotopie rationnelle des espaces fonctionnels. (French. English summary) [The rational homotopy of function spaces] 
Manuscripta Math. 56 (1986), no. 2, 177–191. 


I remember vaguely that one needs a minimal model of $X$ and of $Y$; then a model of the function space is the tensor product of the minimal models.
There are some conditions, which are satisfied for finite dimensional smooth manifolds.
A: A standard reference for this is Hirsch's Differential Topology textbook.  If $X$ is compact near all the topologies you'd like to consider are essentially the same.  Sometimes they're called the Whitney topologies, or the $C^\infty$-topology, there are weak and strong variants that are only relevant when $X$ is non-compact. 
These spaces have the homotopy-type of the space of continuous maps.  The basic idea is to consider $Y$ as a submanifold of some Euclidean space, any continuous map $f : X \to Y$ you can apply a smoothing operator to, then project via the tubular neighbourhood theorem to get a smooth map $X \to Y$ approximating $f$ (in the $C^k$ sense for any $k$ as large as makes sense for your given map $f$), similarly you can apply this construction to families of functions. 
Hirsch doesn't bother to get into the details of the Frechet manifold structure on these mapping spaces but it's available.  Kriegl and Michor's "Convienient setting for global analysis" is a fairly comprehensive (if daunting) reference for this.  But there are other references out there that provide a modest amount of details. 
http://www.ams.org/publications/online-books/surv53-index
We have one of the founders of this subject online -- Richard Palais.  Perhaps he will have some comments eventually. 
edit: Back to your question.  Spaces of continuous maps $C^0(X,Y)$ are a rather traditional thing to study in algebraic topology.  It really depends on what kinds of questions you have about these spaces. For example, if $X$ and $Y$ are Eilenberg-Maclane spaces $C^0(X,Y)$ has much to do with plain old cohomology.  If your spaces $X$ and $Y$ have nice cell decompositions, you can frequently get at aspects of the homotopy-type of $C^0(X,Y)$ via obstruction theory.  But in general these spaces are pretty complicated.  
A: Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent Oberwolfach report from a meeting on the subject. 
If you share what $X,Y$ you are interested in, I may be able to say more, even though I am by no means an expert.  
EDIT: In response to your edit, suppose $\Sigma$ is a compact Riemann surface, $G$ is a compact Lie group, and let's study $k$th homotopy group of $C^0(\Sigma, G)$. Actually, most of the discussion below applies to general $X,Y$ that are, say, compact manifolds or even finite CW-complexes, but let's stick to the case of interest.
There is an well-known homeomorphism  $C^0(S^k,C^0(\Sigma, G))\cong C^0(S^k\times\Sigma, G)$  given by the adjoint, and since any homeomorphism preserves path-components, we can identify $\pi_k(C^0(\Sigma, G))$ with $[S^k\times \Sigma, G]$, the set of homotopy classes of maps from $S^k\times \Sigma$ to $G$. Smashing $S^k\vee\Sigma$ inside $S^k\times\Sigma$ to a point gives $S^k\wedge\Sigma$, which is the $k$-fold suspension $S^k\Sigma$ of $\Sigma$. The cofiber sequence of the quotient map  $S^k\times\Sigma\to S^k\wedge\Sigma$ is an exact sequence, namely,
$$[S^k\Sigma, G]\to  [S^k\times\Sigma, G]\to [S^k\vee \Sigma, G]=\pi_k(G)+[\Sigma, G].$$
Let's suppose that $G$ is simply-connected, so that $[\Sigma, G]$ is a point (as any Lie group has trivial $\pi_2(G)$ and $\Sigma$ is $2$-dimensional). Then the above cofiber sequence is an exact sequence of groups (not just sets). In trying to compute $[S^k\Sigma, G]$ it helps to recall that $G$ is rationally homotopy equivalent to the product of odd-dimensional spheres, so rationally $[S^k\Sigma, G]$ is the product of $[S^k\Sigma, S^m]$'s where $m$ is odd, and of course $[S^k\Sigma, S^m]=[\Sigma, \Omega^kS^m]$. This should allow you to do some computations.
Finally, I wish to comment that the inclusion $C^\infty(X,Y)\to C^0(X,Y)$ is a weak homotopy equivalence i.e. it induces an isomorphism of homotopy groups as any map of a sphere/disk is homotopic to a nearby smooth map. A result of Milnor says that $C^0(X,Y)$ is homotopy equivalent to a CW-complex, provided $X$ is a finite CW-complex (if memory serves me). I am not sure why the same is true for $C^\infty(X,Y)$ so at the moment I do not know if the above inclusion is a homotopy equivalence.
A: First off, you want your source space, $X$, to be compact (technically, sequentially compact will do).  If you don't have that, $C(X,Y)$ need not be locally contractible so no hope of a manifold structure, infinite dimensional or otherwise.  If $X$ is compact, all the arguments for loop spaces follow through, the only thing is that there's a bit more variety in model spaces - the models are spaces of sections of vector bundles over $X$ which, for $X = S^1$, is quite simple but for general $X$ is more variable.  However, all these spaces behave like $C^\infty(X,\mathbb{R}^n)$ for all intents and purposes.
The topology on $C^\infty(X,Y)$ is referred to as the "compact-open" topology, but as I've imposed the restriction that $X$ be compact you could think of it as being a uniform-type of convergence.  Basically, open sets are formed by insisting that all derivatives up to a certain finite order are bounded by a certain bound - the Riemannian structure helps with defining this (though it isn't necessary).  However, I prefer to think of the smooth structure in terms of the exponential map which says that the smooth structure on $C^\infty(X,Y)$ is precisely that which makes a map $Z \to C^\infty(X,Y)$ smooth if and only if $Z \times X \to Y$ is smooth.
As a Riemannian manifold, $C^\infty(X,Y)$ can certainly be given a Riemannian structure.  However, it's a weak structure not a strong one.  I'm not sure about general $X$, but for $X = S^1$ then it's the best that it can be whilst still being weak.
Regarding homotopy types, the inclusion $C^\infty(X,Y) \to C^0(X,Y)$ is a homotopy equivalence.  Ryan's answer contains the idea on how to do that.
However, you're more interested in references.  Ryan's already mentioned the canonical reference, here's a few others:


*

*nLab


*

*Harry's mentioned the page on loop spaces

*http://ncatlab.org/nlab/show/differential+topology+of+mapping+spaces


*my papers (and other stuff)


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*The differential topology of loop spaces

*How to Construct a Dirac Operator in Infinite Dimensions (contains details on types of Riemannian structure in infinite dimensions)

*The Co-Riemannian Structure of Smooth Loop Spaces

*Constructing Smooth Manifolds of Loop Spaces
