Let X be a variety defined by a set of polynomial equations in several variables over a finite field F_q with q elements. In many applications, one is interested in the number of solutions of these equations over finite extensions F_q^r of F_q. One can package these integers together into a generating series in a suitable way to obtain the zeta function of X, which possesses a marvelous structure and is a fundamental object in algebraic geometry.

In the first part of this talk, intended for a general audience, we will introduce the zeta function from scratch, provide several examples, and discuss its properties and applications. In the second part, we will discuss a new result on the computation of zeta functions using Dwork cohomology, and conclude with several interesting combinatorial and geometric implementational issues.