I will report on joint work in progress with Wyss and Ziegler. The first part of this talk will be a gentle introduction to p-adic integration, focusing on the case of quotient singularities where it is closely related to orbifold cohomology. In the second half we will compute p-adic integrals for singular moduli spaces of objects in certain abelian categories (e.g. representations of a symmetric quiver, meromorphic Higgs bundles, etc) and see that p-adic integration can be related to BPS cohomology arising in the theory of Donaldson-Thomas invariants.