Automorphisms of the category of free finitely generated algebras

One of the natural questions of Universal Algebraic Geometry is the following one: when do two algebras from a variety of algebras have the same algebraic geometry?

This question can be interpreted in various ways. For instance, one can say that algebraic geometries of the algebras are the same if the categories of algebraic sets over the given algebras are isomorphic.

An important role in the study of the categories of algebraic sets is played by investigations of automorphisms of the category of free finitely generated  algebras in a given variety.

We will present the method of verbal operations for the study of automorphisms of the category of free finitely generated algebras, consider some results in this area, and discuss open problems.