IN THE first part [5] of this investigation of closed orientable 4-manifolds admitting an action of the 2-dimensional torus group, G = T2, we obtained an equivariant classification of actions with no finite isotropy groups, provided the action was not free. The main tool there was a cross-sectioning theorem [5, 1.12 and 4.41. Our first result in this sequel is to obtain an equivariant classification of all actions. This is accomplished by specifying a cross-section on the boundary of a tubular neighborhood of each orbit with finite isotropy group (E-orbit) and considering the problem of extending this partial cross-section to the rest of the orbit space. This approach is analogous to the equivariant classification of 3-manifolds with S’-action [7] and in principle it goes back to Seifert [8]. We have to consider two cases. If there are points with infinite isotropy group, F u C # @, then there is no obstruction to extending this cross-section and the orbit data form a complete set of invariants. If theaction has only finite isotropy groups, F u C = @, then M is a Seifert manifold and an additional invariant appears, representing the obstruction to extending the cross-section. The remaining sections contain topological results. Since we have an equivariant classification, we may think of our manifolds given in terms of an action and ask when two such manifolds are homeomorphic (diffeomorphic). This problem was solved for simply connected manifolds in our first paper [5; $51. In $2 we investigate the Seifert case, F u C = @, using the techniques of Conner and Raymond [l-4]. In “almost all ” cases we see that two manifolds are homeomorphic if and only if they are equivariantly homeomorphic. This result was obtained independently by H. Zieschang [lo] using different methods. We also observe that M fibers over S’ with fiber a 3-dimensional Seifert manifold. In 93 we show that in the presence of fixed points, F# 0, M can be represented as an equivariant connected sum of “ elementary ” 4-manifolds with G-action. Each elementary manifold has cyclic fundamental group. There are two unfortunate aspects of this result, ’ however. First the mutual homeomorphism relationships of these elementary manifolds are not known. This difficulty resembles the classification problem of lens spaces. Next, the decomposition is not unique and in the light of [5; $5.81 this may turn out to be a rather serious obstacle. $4 is a partial answer to the first problem for manifolds with infinite cyclic fundamental group, i.e. for manifolds with no finite isotropy, E = $3. We call M