As has been discussed in this seminar previously, the (verified) Weil conjectures provide a connection between the number of solutions of a system of polynomial equations over a finite field and the cohomology of the topological space given by the solutions over complex numbers. In this talk we will discuss p-adic integration, which allows us to endow the set of solutions over p-adic integers with a measure, whose volume is related to the number of solutions over the field with p elements, and therefore to the cohomology over the complex numbers. As an application we sketch the proof of Batyrev that two birational projective and smoth CY varieties have the same Betti numbers. If time permits we discuss how this can be extended to a comparison of Hodge numbers, either using arithmetic geometry or motivic integration.