This is an unifying, historical and prospective presentation of my works in homology and homotopy looking it in the general framework of the research in these areas. Starting with the homology of categories, one considers the notions of coefficients giving cohomologies. Extensions, double extensions and multiple extensions are defined and classified by the cohomology. Here comes the link from extensions to homotopy in order to build a theory of extensions of h-objects in categories having homotopy. The last two sections present the needed theories of homotopy for categories and of homotopical algebra.