Many fields of mathematics have their own versions of zeta function: arithmetic and algebraic geometry, dynamical systems, graphs, operator theory. So this entry will take a lot of time to write. The most well-known example is of course the Riemann zeta function.
There are several articles trying to attach zeta functions attached to triangulated categories (and stable model categories…) and study them. Zeta functions express some motivic information, hence the categorical framework must be natural as it is for motives.