Abstract: Introduced by R.T. Powers, $E_0$-semigroups are one-parameter semigroups $\rho=\{\rho_t\}_{t\geq 0}$ of unital normal *-endomorphisms acting on von Neumann algebras, usually the von Neumann algebra $B(H)$ of all bounded linear operators on a separable Hilbert space $H$. $E_0$-semigroups can be regarded as quantum generalizations of the classical time-irreversible dynamical systems, and their study takes into account at a non-commutative level various dissipation mechanisms and state-time evolution phenomena.

It is our purpose, in this presentation, to survey the current state of knowledge of the subject, and to discuss several classification problems.