In the last 15 years many authors have generalized the Banach contraction principle in relation with several classes of generalized contractions. In the present paper we shall establish, in Part I, the inclusion relations between these classes, and a fixed point theorem for spaces with a family of pseudo-metrics, in Part II, some new results on fixed points for multivalued generalized contractions, then in Part III, a fixed point theorem of Krasnoselski type and finally, in Part IV, some fixed point theorems for non-expansive mappings in locally convex spaces.