  We introduce and survey results on two families of zeta functions connected
to the multiplicative and additive theories of integer partitions. In the case
of the multiplicative theory, we provide specialization formulas and results on
the analytic continuations of these "partition zeta functions", find unusual
formulas for the Riemann zeta function, prove identities for multiple zeta
values, and see that some of the formulas allow for $p$-adic interpolation. The
second family we study was anticipated by Manin and makes use of modular forms,
functions which are intimately related to integer partitions by universal
polynomial recurrence relations. We survey recent work on these zeta
polynomials, including the proof of their Riemann Hypothesis.
