When a group of ismorphisms is a Lie group

1. What are the known cases where a group of isomorphisms of a smooth manifolds (diffeomorphisms that respect a given structure on the manifold) is a Lie group? such as: isometries of a compact reimannian manifold, symplectomorphisms of a symplectic manifold...
2. What are the known cases where the diffeomorphism group $\mathrm{Diff}(M)$ of a compact manifold is a Lie group (infinite-dimensional)?