The history of universal algebraic geometry

In this paper, we give a sketch of the evolution of Universal Algebraic Geometry and Logical Geometry. These topics were systematically developed in the works by Plotkin and his followers starting from 1996. A similar theory was constructed by Remeslennikov, Miasniakov and others. We provide the reader with the examples of problems typical to universal algebraic geometry and discuss thoroughly the open problem on isomorphism of isotypic finitely generated groups.