The foundations of measure theory have traditionally uncovered ever deeper infinities. Jan Denef in 1984 reversed this trend in a p-adic setting, using quantifier elimination theorems from model theory to show that integration can be carried out in a restricted, finitary geometric framework. In the 1990's, Batyrev showed how p-adic integration may be used in algebraic geometry, and Kontsevich introduced motivic arc space integration in explanation and extension. Denef-Loeser and Loeser-Cluckers demonstrated the essential identity of these theories, again using model-theoretic results. The talk is intended to describe the above notions, and present a new framework for them. I will mention some simple geometric applications, such as: two smooth projective curves with isomorphic complements, are isomorphic; and attempt to explain how logic can bring out such facts. Based on joint work with David Kazhdan.