Supercurves are the simplest class of complex super manifolds, "one-dimensional" in some sense and thus analogs of Riemann surfaces. I will describe a remarkable duality between pairs of super curves that generalizes Serre duality for Riemann surfaces. Self-dual super curves are precisely the "super Riemann surfaces" introduced by physicists in connection with string theory. I'll suggest connections between this duality and the classical duality between points and hyperplanes in projective space. No prior knowledge of supergeometry is required.