During the academic year 1987–8 a group of young mathematicians at the University of Bath prepared (for the first time) a pamphlet, Master of Science in nonlinear mathematics, that contained the following entry.

PG14 Symmetry and the Maximum Principle

Naturally, the pamphlet did not state how this goal was to be reached in twenty lectures to students who could not be assumed to have any experience whatever of partial differential equations. Nor were detailed suggestions issued to me when, in the autumn of 1988, I joined the University of Bath and was ordered to give these lectures. What the authors of the pamphlet did do, however, was to attend the lectures themselves, to ask awkward questions, to imbue the course PG14 with their own youthful verve, and to appeal to my vanity by suggesting that I prepare something like the present book.

This explanation should indicate that the word Introduction in the title of the book is no gloss. I offer genuine apologies to B. Gidas, W.-M. Ni and L. Nirenberg for the extent to which I have used their paper Symmetry and related properties via the maximum principle (1979), to H. Berestycki and L. Nirenberg for my use of the easiest part of On the method of moving planes and the sliding method (1991), and to D. Gilbarg and N.S.