The topic I shall discuss is mathematics and rheology and I shall, of course, concentrate on those aspects of mathematical rheology in which I have been interested. Many of you will no doubt recall that Sir Geoffrey Taylor chose a similar topic as the subject for his Presidential Address to the International Conference on Rheology held at Oxford in 1953. In it he advanced what seemed to me a rather pessimistic appraisal of the role of mathematics in rheology. I shall try to be more optimistic. Unfortunately, I was not able to attend the Oxford conference myself and merely read Sir Geoffrey's talk in the proceedings of the conference. I conjecture from its pessimistic note that Sir Geoffrey must have been talking before dinner. I like to think that if he had been talking after dinner, he too would have been more optimistic. In his preprandial pessimism he took the view that the mathematician must at best have the role of a handmaiden to the experimental rheologist, solving—or more often failing to solve—particular little problems that are posed. I would much rather regard his role as one of equal responsibility with the experimental rheologist for the advance of the subject—at times running ahead of the experimenter and determining the pattern of progress, at other times lagging behind.