The most important thing that a young mathematician needs to learn is of course mathematics. However, it can also be very valuable to learn from the experiences of other mathematicians. The ﬁve contributors to this article were asked to draw on their experiences of mathematical life and research, and to oﬀer advice that they might have liked to receive when they were just setting out on their careers. (The title of this entry is a nod to Sir Peter Medawar’s well-known book, Advice to a Young Scientist.)

The resulting contributions were every bit as interesting as we had expected; what was more surprising was that there was remarkably little overlap between the contributions. So here they are, ﬁve gems intended for young mathematicians but surely destined to be read and enjoyed by mathematicians of all ages.