Rings satisfying a polynomial identity have been proven to enjoy important properties. The pioneering works of Jacobson, Kaplansky and Levitzki resulted in a solution to the bounded Kurosh problem stating that a finitely generated associative algebra over a field in which every element satisfies an algebraic equation of bounded degree is finite dimensional. The Kurosh problem is an analogue of the bounded Burnside problem for groups for which Zelmanov was awarded the Fields medal.
The work of Kemer on $T$-deals and its application to yield a positive answer to the Specht conjecture has been on the spotlight and to some extent under scrutiny in the past few years. I will review some of the recent developments and activities in this regard. Then I will review the results about group rings and enveloping algebras that satisfy a non-matrix polynomial identity.