The Jacobian of any compact Riemann surface carries a natural theta divisor, which can be defined as the zero locus of an explicit function, the Riemann theta function. I will describe a generalization of this idea, which starts by replacing the Jacobian with the moduli space of sheaves over a Riemann surface or a higher dimensional base. These moduli spaces also carry theta divisors, described as zero loci of "generalized" theta functions. I will discuss recent progress in the study of generalized theta functions. In particular, I will emphasize an unexpected geometric duality between spaces of generalized theta functions, as well as its geometric consequences for the study of the moduli spaces of sheaves.