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Geological settings with reservoir characteristics include fractures with different material and geometrical properties. Hence, numerical simulations in applied geophysics demands for computational frameworks which efficiently allow to integrate various fracture geometries in a porous medium matrix. This study presents a modeling approach for single-phase flow in fractured porous media and its application to different types of non-conforming mesh models. We propose a combination of the Lagrange multiplier method with variational transfer to allow for complex non-conforming geometries as well as hybrid- and equi-dimensional models and discretizations of flow through fractured porous media. The variational transfer is based on the $L^2$-projection and enables an accurate and highly efficient parallel projection of fields between non-conforming meshes (e.g.,\ between fracture and porous matrix domain). We present the different techniques as a unified mathematical framework with a practical perspective. By means of numerical examples we discuss both, performance and applicability of the particular strategies. Comparisons of finite element simulation results to widely adopted 2D benchmark cases show good agreement and the dual Lagrange multiplier spaces show good performance. In an extension to 3D fracture networks, we first provide complementary results to a recently developed benchmark case, before we explore a complex scenario which leverages the different types of fracture meshes. Complex and highly conductive fracture networks are found more suitable in combination with embedded hybrid-dimensional fractures. However, thick and blocking fractures are better approximated by equi-dimensional embedded fractures and the equi-dimensional mortar method, respectively.