Group and Polynomial Identities in Group Rings

In the 1980s Brian Hartley conjectured that if the unit group, U(FG), of a torsion group ring FG satisfies a group identity, then FG satisfies a polynomial identity. The aim of this survey is to review the most relevant results that arose from the proof of this conjecture and discuss some recent developments and open questions concerning ∗-group identities for U(FG) and group identities for the subgroup of its unitary units.