In algebraic geometry there is a definition for the genus of any algebraic curve C. When the field of definition for C is the complex numbers, and C has no singular points, then that definition coincides with the topological definition applied to the Riemann surface of C (its manifold of complex points). The definition of elliptic curve from algebraic geometry is non-singular curve of genus 1.
in graph theory, the genus of a graph is a integer n such that the graph can be drawn without crossing itself on a surface with n-handles. Thus, a planar graph has genus 0. (can be drawn on a sphere without self-crossing)