The classification of simply-connected 5-manifolds by Smale-Barden is a classical result in manifold topology.In this talk I will report results on the classification of 5-manifolds with fundamental group Z2 and Z.The machinery is the modified surgery method of Kreck.These results are related to interesting problems on group actions on simply-connected 5-manifolds and self-diffeomorphisms of smooth 4-manifolds.