Current algebras on Riemann surfaces

We will consider three known classes of Lie algebras of meromorphic currents on Riemann surfaces. These are the loop algebras, also known as Kac–Moody algebras, and two subsequent generalizations of them, the Krichever–Novikov algebras and Lax operator algebras. We will present known facts of the structure theory for Lie algebras of those classes, as well as their applications, mainly to mathematical physics. In most details, we are going to consider Lax operator algebras having recently appeared in the works of I. Krichever and the author, as well as their applications to theory of finite-zone integration. We would like mention that those algebras are defined in terms of A. Tyurin parameters of holomorphic vector bundles on Riemann surfaces.