The mathematical theory of Gaussian multiplicative chaos was founded by J.P. Kahane in 1985. This theory has numerous applications in mathematical physics: 2d Liouville quantum gravity in the conformal gauge (boundary and non boundary Liouville measure, KPZ equation), the Kolmogorov-Obukhov model of energy dissipation in 3d turbulence, the maximum of log-correlated fields, etc… par In this talk, we will review the theory of Kahane and discuss numerous extensions that have been developed the past few years, in particular to the complex case. Special emphasis will be made on applications which motivate these extensions. This is based on joint works with B. Duplantier, H. Lacoin, T. Madaule, R. Rhodes, S. Sheffield.

Dépt. de Mathématique (DMA), Ecole Normale Supérieure Paris