Professor Dancer has made fundamental contributions to the theory of partial differential equations, non-linear analysis, and applications. His work has led to new areas of research in topics such as global bifurcation, symmetry breaking, degree theory, and infinite dimensional dynamical systems, in many cases introducing concepts from one area into another. He only works on difficult problems of some consequence, often motivated by and having consequences for physical and biological phenomena, such as catalysis theory and population systems with large interactions.