Finite basis problem for varieties of algebraic systems

This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with the property that they and their subvarieties are finitely based. A special attention is paid on the varieties of semigroups, groups, associative, Lie and other nonassociative algebras.