In　this　paper,　we　present　a　discontinuous　Galerkin　(DG)　method　for　the　simulation　of　incompressible　and　immiscible　two-phase　flow　in　fractured　porous　media　with　capillary　pressure.　The　hybrid-dimensional　fracture　models　are　considered　in　which　fractures　are　treated　as　(d　-　1)-dimensional　interfaces　in　the　d-dimensional　domain　and　fluid　exchange　between　fractures　and　surrounding　matrix　is　taken　into　account.　The　approximation　schemes　combining　the　DG　method　with　the　fully　implicit　backward　Euler　time　discretization　are　formulated　for　models　with　a　single　fracture　and　intersecting　fractures　network,　and　the　stability　and　unique　solvability　are　established.　Optimal　error　estimates　in　the　discrete　H1-norm　for　saturation　and　pressure　are　derived.　Numerical　experiments　in　two-dimensional　geometry　are　conducted　to　verify　the　accuracy　of　our　theoretical　analysis.　In　addition　to　the　backward　Euler　formulation,　we　also　implement　the　algorithm　with　a　kind　of　conventional　IMplicit　Pressure　Explicit　Saturation　(IMPES)　method.　&　COPY;　2023　Elsevier　Inc.　All　rights　reserved.