Zeta functions are central topics in number theory and arithmetic algebraic geometry. They arise in many different forms. They can be viewed as generating functions for counting "points"
(or "solutions") of polynomial equations, thus contain deep arithmetic and geometric information of the equations. In this lecture, we will explain various zeta functions from this point of view, including the Riemann zeta function, the Hasse-Weil zeta function, and zeta
functions over finite fields.