Many fields of mathematics have their own versions of zeta functions: arithmetic geometry and algebraic geometry, dynamical systems, graphs, operator algebra. 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.
multiple zeta values, motivic multiple zeta values, motivic integration, motive
there are attempts to understand the Riemann zeta function as the spectrum of a Hamiltonian of a quantum mechanical system. See at Riemann hypothesis and physics.
A general review is in
Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228:4 (1995) 121–163, and preprint MPIM1992-50 pdf
Nobushige Kurokawa, Zeta functions over $F_1$, Proc. Japan Acad. Ser. A Math. Sci. 81:10 (2005) 180-184 euclid
M. Larsen?, V. A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3, 1 (2003) 85–95; Rationality criteria for motivic zeta functions, Compos. Math. 140:6 (2004) 1537–1560
Vladimir Guletskii, Zeta functions in triangulated categories, Mathematical Notes 87, 3 (2010) 369–381, math/0605040
M. Kontsevich, Notes on motives in finite characteristics, math.AG/0702206