Igusa's local zeta function of a polynomial f over the p-adic integers is related to the number of solutions of f(x) $\equiv $ 0 mod p$^\mathrm {n}$ and to exponential sums mod p$^\mathrm {n}$. There are several conjectures, one of which relates the poles of this zeta function to the local monodromy of the singularities of f = 0 . Results of Igusa, Loeser, Meuser, Veys, Denef and others, indicate several connections with topology and singularity theory. We will also focus on the special case of curves and of relative invariants of reductive groups. Integration over p-adic subanalytic sets and its applications will be discussed briefly.