We can think of diffeomorphisms as causing a transformation on a manifold. From that point of view, it is natural to consider two diffeomorphisms as equivalent when there is a change of variables that reduces one to the other. We will study what obstructions to equivalence appear and how can one show that related diffeomorphisms are equivalent under smooth changes of variables. Since this is a "natural" problem, we will also discuss some applications to other fields such as Riemannian geometry.