A complex supercurve is a complex supermanifold of dimension 1—1, or more precisely a family of these including odd parameters. These objects include, but are more general than, the super Riemann surfaces (a.k.a superconformal manifolds) that appear in string theory. I will review how classical facts about Riemann surfaces generalize to the context of supercurves, emphasizing the gaps and open problems in the theory. Topics include divisors, the dual of a supercurve (the self-dual ones are precisely the super Riemann surfaces), Abel’s Theorem and the Jacobian of a supercurve, Abelian supervarieties and theta functions, and super Poincare series