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The classification of Seifert manifolds was given in terms of numeric data bySeifert in 1933, and then generalized by Orlik and Raymond in 1968 to circleactions on closed 3d manifolds. In this paper, we further generalize theclassification to circle actions on 3d manifolds with boundaries by adding anumeric parameter and a union of cycle graphs. Then we describe the equivariantcohomology of 3d manifolds with circle actions in terms of ring, module andvector-space structures. We also compute equivariant Betti numbers andPoincar\'{e} series for these manifolds and discuss the equivariant formality

Classification and equivariant cohomology of circle actions on 3d manifolds