The P=W conjecture: statement, motivation, known cases

This is a survey talk explaining the statement of the so called P=W conjecture (due to M. A. de Cataldo, T. Hausel and myself), its motivation, stemming from the determination, due to T. Hausel and F. Rodriguez-Villegas, of the Mixed Hodge polynomial of some character varieties, the few cases in which this conjecture was proved and the problems one finds to extend these proofs to the general case.