Groups of automorphisms of $d$-adic rooted trees  have been studied for years as an important source of groups with interesting properties. For example, many of their subgroups constitute a counterexample to the General Burnside Problem. The question whether every Engel group is locally nilpotent is the analogue of the general Burnside problem in the realm of Engel groups.

In this talk we survey recent results about Engel conditions in some general families of groups of automorphisms of rooted trees.

This is joint work with G.A. Fern\'andez-Alcober, A. Garreta, and G. Tracey.