If M is a non-compact manifold, then Diff(M) is not a manifold. Yet it still has manifold-like properties. What is it?

Question

If $M$ is a non-compact smooth manifold, then $\text{Diff}(M)$ is not locally compact and hence not a manifold. Yet it still has manifold-like properties. In fact it has Lie group-like properties. It's "smooth" and a group. Yet it seems to be so big that not even a infinite-dimensional Lie group can capture its essence. What is it exactly?