Congruence classes of varieties of algebras

The congruence class of a variety of algebras is the class of all isomorphic copies of congruences lattices of algebras in the variety. Most of this talk resolves around the following two questions: Is it possible to characterize lattices in a given congruence class? To which extent a variety of algebras is determines by its congruence class? The first question seems very difficult. It is even difficult, for a given class, to prove that a will-to-be characterization is not valid. However, there is an efficient tool that allows to distinguish congruence classes, based on poset-indexed diagrams of algebras. I shall illustrate it with several examples.