In this talk I will discuss a new alternative for finitely generated amenable groups, first established together with H. Izeki for Liouville groups and then extended to the amenable case in a forthcoming joint work with H. Izeki, R. Ji and Y. Wu. Given a finitely generated amenable group, either every action on a finite-dimensional complete CAT(0)-space has a fixed point or the group has an action on a Euclidean space without a fixed point. The alternative implies that amenable torsion groups and simple amenable groups (such as important classes of groups constructed by Grigorchuk, Juschenko-Monod, Matte Bon, Nekrashevych, and others) cannot act on any finite-dimensional CAT(0)-space without having a global fixed point.