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 bundles (sheaves) over a Riemann surface (or a higher dimensional base). These moduli spaces also carry theta divisors, described via "generalized" theta functions. In this talk, I will describe recent progress in the study of generalized theta functions.