This paper reports on some results concerning:a) The homotopy type of the group of diffeomorphisms $\mathrm{Diff}\left(M^n\right)$ of a differentiable compact manifold $M^n$ (with $C^{\infty }$-topology).b) the result of the homotopy comparison of this space with the group of all homeomorphisms Homeo $M^n$ (with $C^o$-topology). As a biproduct, one gets new facts about the homotopy groups of $\mathrm{Diff}(D^n,\partial D^n),{\mathrm{Top}}_n$, ${\mathrm{Top}}_n/O_n$ and about the number of connected components of the space of topological and combinatorial pseudoisotopies.The results are contained in Section 1 and Section...