Hurwitz curves

The present paper is a survey of recent results about Hurwitz curves, their braid monodromy invariants, and their applications to $H$-isotopy and regular homotopy problems. The second part of the survey is devoted to a discussion of the applicability of braid monodromy invariants of branch curves for generic coverings of the projective plane as invariants distinguishing connected components of the moduli space of algebraic surfaces (in the algebraic case) and distinguishing symplectic structures on four-dimensional varieties (in the symplectic case).