About monodromy and quantum cohomology: while everybody knows that Gromov-Witten invariants are invariant under deformations, the immediate consequence that they are invariant under the monodromy action for any smooth family of varieties is sometimes forgotten. In other words, they are invariant under the mapping class group. This can be extremely useful in computations when the mapping class group is big.
To elaborate a bit: when we are saying "GW invariants are invariant under small deformations", then we are implicitly already using the Gauss-Manin connection: If $X \to T$ is a family, say over the disc, then it doesn't make sense to identify Gromov-Witten invariants of $X_{t_1}$ and $X_{t_2}$ unless you know which GW-invariants to compare; in other words, given cohomology classes $\gamma_1, \dots, \gamma_n$ and a homology class $\beta$ on $X_{t_1}$, we need to find corresponding classes on $X_{t_2}$. Well, fortunately, this is not a problem, as $X_{t_1} \cong X_{t_2}$ as smooth manifolds, and thus $H^*(X_{t_1}) \cong H^*(X_{t_2})$. This identification is nothing but the Gauss-Manin connection. In particular, when the bases $T$ is more complicated, you need to choose a path from $t_1$ to $t_2$ to obtain the identification; and it will depend on the homotopy class in the path. In particular, when $T$ is not simply-connected, we get a representation of $\pi_1(T)$ on $H^*(X_0)$, i.e. the monodromy action. Since it is pieced together out of identifications $H^*(X_0) \cong H^*(X_{t_1}) \cong H^*(X_{t_2}) \cong \dots$ as above, GW-invariants are invariant under this group action:
$ \langle \gamma_1, \dots, \gamma_n \rangle_{\beta}^{g,n} = \langle \Phi(\gamma_1), \dots, \Phi(\gamma_n) \rangle_{\Phi(\beta)}^{g,n} $
for any $\Phi$ in the image of $\pi_1(T) \to \mathrm{Aut} H^*(X, \mathbb{Q})$. (The mapping class group is basically the biggest possible group $\pi_1(T)$ you can obtain this way, i.e. the fundamental group of the moduli space of varieties diffeomorphic to $X$.)