This paper reviews the main probabilistic results on multivariate extremes. Historically, this branch of probability focussed on the limiting distribution of the componentwise maximum of independent and identically distributed random vectors. A number of equivalent asymptotic characterizations are available and will be summarized. Inferential issues arising from the statistical application of classical results are discussed. They motivate the development of alternative models for multivariate extremes, based on multivariate threshold exceedances, which have received increasing attention in the literature and will also be reviewed.