In this work, we extend the single-phase Darcy flow model proposed in [25], [12] to two-phase flow. We propose two models for two-phase Darcy flow through fracture networks in porous media, in which the (d − 1)-dimensional flow in the fractures is coupled with the d-dimensional flow in the matrix, leading to the so called hybrid-dimensional Darcy flow models. They both account for fractures acting either as drains or as barriers, since they allow pressure jumps at the matrix-fracture interfaces. The models also permits to treat gravity dominated flow as well as discontinuous capillary pressure at the material interfaces. We adapt the Vertex Approximate Gradient (VAG) scheme to this problem, in order to account for anisotropy and heterogeneity aspects as well as for applicability on general meshes. Several test cases are presented to compare our hybrid-dimensional models to the hybrid-dimensional, continuous pressure model (proposed in [9]) and to the generic equi-dimensional model, in which fractures have the same dimension as the matrix. This does not only provide quantitative evidence about computational gain, but also leads to deep insight about the quality of the proposed reduced models.