Replicator dynamics: Old and new

We introduce the unilateral version associated to the replicator dynamics and describe its connection to on-line learning procedures, in particular to the multiplicative weight algorithm. We show the interest of handling simultaneously discrete and continuous time analysis. We then survey recent results on extensions of this dynamics as maximization of the cumulative outcome with alternative regularization functions and variable weights. This includes no regret algorithms, time average version and link to best reply dynamics in two person games, application to equilibria and variational inequalities, convergence properties in potential and dissipative games.