In this talk we will discuss the theory of zeta integrals of smooth representations of GL(n) of a p-adic field. These integrals are essentially Mellin transforms of locally constant functions on the group. We will outline the roots of the subject, beginning with Tate's 1950 thesis, and describe how the seemingly analytic framework can be interpreted purely algebraically, concluding with a description of recent results generalizing the theory to work over rings.