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.