Zeta functions are objects that arise in many areas of mathematics. This talk will be about the Dedekind zeta function of a number field and the Hasse-Weil zeta function of a smooth curve over a finite field; the goal is to explain how these zeta functions contain (respectively) arithmetic and geometric information. If time permits, I will also talk about the zeta functions of schemes that are of finite type over the integers. These generalize the previous types of zeta functions.