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The present authors have introduced polynomial splines over T-meshes (PHT-splines) and provided the theories and applications for PHT-splines over hierarchical T-meshes. This paper generalizes PHT-splines to arbitrary topology over general T-meshes with any structures. The general PHT-spline surfaces can be constructed through an unified scheme to interpolate the local geometric information at the basis vertices of the T-mesh. We also discuss the edge insertion and removal algorithms for PHT-splines over general T-meshes. As applications, we present algorithms to construct a spline surface over a T-mesh from a quadrilateral mesh.