In 1902, William Burnside wrote: "A still undecided point in the theory of discontinuous groups is whether the order of a group many not be finite while the order of every operation it contains is finite." Since then, the Burnside problem, in different guises, has inspired a considerable amount of research. One variant of the Burnside problem, the restricted Burnside problem, asks whether (for a given r and n) there is a bound on the orders of finite r-generator groups of exponent n. This book provides the first comprehensive account of the many recent results in this area. By making extensive use of Lie ring techniques it allows a uniform treatment of the field and includes Kostrikin's theorem for groups of prime exponent as well as detailed information on groups of small (3,4,5,6,7,8,9) exponent. The treatment is intended to be self-contained and as such will be an invaluable introduction for postgraduate students and research workers. Included are extensive details of the use of computer algebra to verify computations.