The topic of growth entered group theory, with a geometric motivation, at the middle of last century. It associates to each finitely generated group a number-theoretical function, its growth, and investigates the relationship between the properties of the group and of its growth function. The subject has attracted attention gradually, until, in about 1980, two seminal theorems were proved: first Misha Gromov determined the groups of polynomial growth, and a short time later Slava Grigorchuk constructed groups of intermediate growth. Both theorems were the starting points for rich mathematical developments. As far as I know, there is no detailed treatment of growth in book form. Of course, I should not ignore Pierre de la Harpe's remarkable text Topics in Geometric Group Theory, a large part of which is devoted to growth, but in that text most results are quoted without proofs (an exception is the chapter about Grigorchuk's group, but even about that group we were able to include in the present text more recent results).

For several years now, I have been teaching courses devoted to one or the other of the above results. Most of these were one-semester courses in the Einstein Institute of Mathematics in the Hebrew University, but some were given at several Italian universities. The present notes were prepared for, and based on, these courses. They can be divided roughly into four main parts: introductory, polynomial growth, intermediate growth, and miscellany.