Trevor Wooley is renowned for his deep contributions to analytic number theory, and especially additive number theory. By a penetrating analysis of classical methods, combined with the development of incisive new ideas, he has been able to make important progress on a series of central problems, with far-reaching consequences. In particular, he has contributed significantly to our understanding of the Vinogradov mean value theorem, Waring’s problem, the theory of exponential sums, diophantine approximation, the validity of the local-to-global principle and the non-trivial integral zeros of systems of forms in many variables.