 A locally finite variety is one whose finitely generated free algebras are finite, such as pointed sets, graphs (whether of the directed or undirected variety), and distributive lattices. A congruence permutable variety is a variety all of whose algebras are congruence permutable. Maltsev characterized these in terms of a necessary and sufficient condition on their theories, namely that \(F\)(3) contain an operation \(t(x, y, z)\) for which \(t(x, x, y) = t(y, x, x) = y\) are in the theory. Analogous notions are congruence distributivity and congruence modularity, for which there exist analogous syntactic characterizations of varieties of algebras with these properties. A more recently developed power tool for this area is McKenzie’s notion of tame congruences, facilitating the study of the structure of finite algebras.