n-Engel group is a group that satisfies a group law $[x, y, \ldots y] = 1$, where $y$ is repeated $n$ times. The following natural question was asked in the 1950s: if such a group is locally nilpotent. For $n \leq 3$ the answer is affirmative. For lager $n$ this question seems to be quite delicate. Namely, if one adds some extra restrictions together with the n-Engel identity, the resulting group is indeed locally nilpotent. For example, finitely generated n-Engel groups that are at the same time residually finite, or solvable are nilpotent. However, we expect that finitely generated free n-Engel groups are not nilpotent for sufficiently large $n$.
The Engel problem has a connection with the Burnside problem, which asks whether a finitely generated group with a group law $x^n = 1$ is necessarily finite. The new proof of the Burnside problem that was recently obtained by E. Rips, K. Tent and myself helps to make progress in the Engel problem. I will talk about my current progress and difficulties along this way.