Given two topological spaces $X, Y$, the only example I know of a topology on the space $\mathcal C(X,Y)$ of continuous mappings from $X$ to $Y$ is the compact-open topology. However I presume that there are other interesting topologies as well, which are useful in other situations. What are some examples, and what is a most interesting situation for its use?
In particular, is there any particular interesting topology if $X$, $Y$ are both smooth manifolds and we are considering differentiable maps instead of continuous maps?