Minimal varieties of associative PI (super)-algebras with respect to their (graded) exponent

Giambruno and Zaicev (Trans Am Math Soc 355: 5091-5117, 2003) characterized varieties of associative PI-algebras over a field of characteristic zero which are minimal of fixed exponent. The aim of the present survey paper is to discuss this result and the recent developments on the corresponding problem for PI-algebras endowed with a Z_2-grading. In particular, we provide an example of minimal superalgebra not generating a minimal supervariety of fixed superexponent, thereby partially answering the question of which actually are the generators of minimal supervarieties of finite basic rank.