Arnold's Seminar Feb 23, 1999 S.Natanson HURWITZ SPACES Those are the spaces of pairs ( P, f), where P is a Riemannian surface and f is a meromorphic function on it. The simplest example -- space of polynomials of a given degree. These spaces being studied since the works of Hurwitz ( more than a century ago) have a number of remarkable topological and analytic properties. Recently there arised a deep relation with topological field theory. The talk will contain both classical and new results in this subject.