Professor Trudinger has earned an international reputation for his research and scholarship in the study of existence of solutions to partial differential equations. A priori estimates for Hölder continuity and the gradient play an important part in proving the existence of solutions in the non-linear case. Professor Trudinger's published papers illustrate his great skill in devising special methods for making such estimates. His papers also show that he has considerable ability to handle other approaches and obtain results for a wide range of problems.