In this chapter we develop the theory of local zeta functions. Specifically, given a local field k, we make compatible choices of Haar measures for the additive and multiplicative group of k, and we discuss Fourier analysis on these groups. We further show that local zeta functions, defined in terms of Fourier analysis, satisfy a very general functional equation, which we describe in detail for all the cases of interest, namely when k is isomorphic to \(\mathbb {R}\) , \(\mathbb {C}\) , or a p-adic field.