Let F be a field and G a group. We consider the unit group, \(\mathcal {U}(FG)\) , and ask when it will satisfy a group identity \(w(x_1,\ldots,x_n)=1\) . We present the proof of Hartley’s conjecture, namely that if \(\mathcal {U}(FG)\) satisfies a group identity, and G is torsion, then FG satisfies a polynomial identity. We then prove necessary and sufficient conditions for \(\mathcal {U}(FG)\) to satisfy a group identity. (When G is not torsion, we must assume that G/T is a u.p. group for the sufficiency, where T is the set of torsion elements of G.)