Summary: The authors discuss the existence of solutions to evolution quasivariational inequalities: Find \(u(t)\in C(u(t)): \langle u(t)+ f(t)\), \(v- u'(t)\rangle\geq 0\) for all \(v\in C(u(t))\), with Lipschitz-continuous dependence \(u\to C(u)\). These problems are special cases of the sweeping processes \(-u(t)\in N_{C(t,u(t))}(u'(t))\), where \((t,u)\to C(t,u)\) is Lipschitz continuous. The authors prove the existence of solutions to both types of inclusions.