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We study ramified covers of the projective plane. Given a smooth projectivesurface S and a generic enough projection of S to the projective plane, we geta cover of the plane ramified over a plane curve. The branch curve is usuallysingular, but is classically known to have only cusps and nodes assingularities for a generic projection.  Several questions arise. First, what is the_geography_ of branch curves amongall nodal-cuspidal curves? Second: what is the_geometry_ of branch curves? Inother words, how can one distinguish a branch curve from a non-branch curvewith the same numerical invariants? For example, a plane sextic curve with sixcusps is known to be a branch curve of a generic projection iff its six cuspslie on a conic curve, i.e., form a special 0-cycle on the plane.  We start with reviewing what is known about the answers to these twoquestions, mentioning both simple and some non-trivial results. We continuewith study of classical work of Beniamino Segre which gives a complete answerto the second question in the case when S is a smooth surface in athree-dimensional projective space. We give an interpretation of Segre's workin terms of a study of Picard group of 0-cycles on a singular plane curve B.  We also review examples of small degree.  The Appendix written by Eugenii Shustin shows the existence of many newZariski pairs of plane curves.  We hope to continue this paper with a generalization to the case of smoothsurfaces in a projective space of any dimension.

On ramified covers of the projective plane I: Segre's theory and classification in small degrees, with Appendix by Eugenii Shustin