Résume | The theory of motives, introduced in the sixties, studies the common properties of the different cohomology theories (de Rham, Betti, etale, crystalline, etc) of algebraic varieties. In the same vein, the theory of noncommutative motives, introduced much more recently, studies the common properties of the different invariants (K-theory, cyclic homology, topological Hochschild homology, etc) of "noncommutative algebraic varieties". The bridge from the former theory to the latter consists of the passage from an algebraic variety to its derived category. The aim of this talk, prepared for a broad audience, is to give an overview of the theory of noncommutative motives and to describe some of its manyfold applications to adjacent areas of mathematics. |