$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Imb{Imb}$Notation: Let $M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $\Diff(M)$ the group of diffeomorphisms of $M$ and $\Imb(S, M)$ the space of smooth imbeddings of $S$ into $M$.

A classical result of R. Palais from the 1960 paper *Local triviality of the restriction map for embeddings* says that the map $\Diff(M)\rightarrow\Imb(S, M)$ given by restriction is a fibration.

I feel like I've heard during numerous teas that there are various refinements and generalizations of this due to J. Cerf and (possibly) others.

(1) Can anyone summarize what else is known in this direction beyond the theorem of Palais?

(2) Is there a way to see Palais' result easily? [*added:* from the responses it sounds like the original paper is still a great way to see this result. But see the answers of Randal-Williams and Palais for an alternate route.]