The restricted Burnside problem asks whether, for each natural numbers r and n, there are only finitely many finite r-generated groups of exponent n. The solution of this problem was given by Kostrikin in the 1960s for prime exponent, then by Efim Zelmanov in 1991, for which he was awarded the Fields medal in 1994. In fact, both Kostrikin and Zelmanov results concern Lie algebras, and are a perfect illustration of Lie methods in group theory: how to reduce questions on groups to questions on Lie algebras. Starting from a finitely generated group, one may construct an "associated Lie algebra" which, for the case of exponent p, is n-Engel, i.e. satisfies the n-Engel identity: [x,y,y,...,y] = 0 (n times). For that case, the restricted Burnside problem reduces to proving that every finitely generated n-Engel Lie algebra is nilpotent.

In 1988, Zelmanov proved the ultimate generalization of Engel's classical result: every n-Engel Lie algebra over a field of characteristic 0 is nilpotent. This theorem has the following consequence: for every n there exists N such that every n-Engel Lie algebra of characteristic p>N is nilpotent. It also has consequences for Engel groups.

The proof is rather involved and consists mainly of some intense Lie algebra computations, sprinkled with several beautiful tricks. In particular, the surprising use of the representation theory of the symmetric group has inspired several other authors since then.

In this talk, I will present a little bit of all this. For instance, we will study the case of 3-Engel Lie algebras and I will explain how some part of Zelmanov's proof was re-used by Vaughan-Lee and Traustason to reduce the algorithmic complexity of computing in 4-Engel Lie algebras.