Some new parallels between groups and Lie algebras, or What can be simpler than multiplication table?

I shall give a survey of some recent developments in the study of equations in groups and Lie algebras. We shall discuss several related invariants, such as commutator width of groups and bracket width of Lie algebras, focusing on parallels between the two algebraic structures. If time permits, we shall also consider some local-global invariants, particularly Tate--Shafarevich groups and algebras.