He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan.

The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his.

He later worked with Jacquet on the zeta function of a simple algebra.

His book Topologie Algébrique et Théorie des Faisceaux from 1958 was, as he said, a very unoriginal idea for the time (that is, to write an exposition of sheaf theory); as a non-specialist, he managed to write an enduring classic.

He also wrote texts on Lie groups, abstract algebra and mathematical analysis.