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.