After the introduction of the (first) Grigorchuk group, that was the first example of a group with intermediate growth, automata groups became a very popular topic in geometric group theory, giving rise to many examples of groups with special and exotic properties and showing interesting and surprising connections with other branches of Mathematics.

In this talk, after a gentle introduction to the topic, I will present a list of open problems and conjectures concerning the algebraic structure and the boundary action of automata groups.
In particular I will focus my attention on some aspects (freeness, duality approach, essentially/ totally (non-) free actions) studied in collaboration with E. Rodaro.