### Abstract

Twisted K-theory has been invented by P. Donovan and the author more than 35 years ago. The motivation at that time was to give a satisfactory Thom isomorphism (and Poincaré duality theorem) in K-theory.

In this talk, we revisit the subject in the light of new developments (J. Rosenberg, M.-F. Atiyah and G. Segal...). We also sketch a proof of the Thom isomorphism theorem in this setting and give explicit computations in the equivariant case, in relation with the "Chern character" for finite groups introduced by Baum, Connes and Slominska.