The last decade has seen a strong increase of research into flows in fractured porous media, mainly related to subsurface processes but also in materials science and biological applications, as connected fractures can totally dominate flow patterns. Due to the fractures' characteristics as approximately planar discontinuities with an extreme size-to-width ratio, they challenge standard macroscale mathematical and numerical modeling of flow based on averaging. Thus, over the last decades, various, and also fundamentally different, modeling approaches have been developed. This paper reviews common conceptual models and discretization approaches for flow in fractured porous media, with an emphasis on the dominating effects the fractures have on flow processes. In this context, the paper discusses the tight connection between physical and mathematical modeling and simulation approaches. Extensions and research challenges related to transport, multi-phase flow and fluid-solid interaction are also commented on.