A vast variety of physical phenomena contain thin features or structures which play a dominant role within the process. Examples of such processes include, but are not limited to flow in fractured media, mechanics of composite materials and biological root systems, and the effects of extraction wells in the subsurface. Modeling these features as lower-dimensional domains within a numerical scheme has proven to be an effective strategy. This minisymposium focuses on recent developments in discretization schemes tailored to solve complex problems with embedded features while respecting physical conservation laws.