Abstract: The categorical group of equivalences. The homotopy category and the group of homotopy classes of homotopy self equivalences of a space. General properties of this group: finite presentation, rank, nilpotency of subgroups. Variations: fiber homotopy equivalences, equivariant homotopy equivalences, H-equivalences, stable homotopy equivalences. Localizations and computations for specific spaces. Applications to the classification of fiber spaces, spaces of the same n-type, homotopy actions of a group on a space.