Professor Mahony has made significant contributions over a wide range of applied mathematics, demonstrating not only an ability to extract the important physical mechanisms in highly complex non-linear phenomena, but also to make a careful analysis of the mathematical subtleties involved, and often to expose and identify hidden flaws which have hitherto been unnoticed. He has made distinguished contributions to the theory of supersonic flow and shock waves, the bending of thin plates, the theory of matched expansions and, most recently, to the non-linear propagation of long waves including the theory of cross waves.