We discuss mathematical modeling and analysis of the incompressible viscous flow at the interface of permeable media. Very recently, a simplified theory with asymptotic modeling and related approximations was extensively developed by [Angot et al., 2017] to provide physically relevant jump interface conditions for the two- or three-dimensional non-inertial flow at the interface of a permeable medium. The results of well-posedness analysis for three problems issued from this asymptotic model and covering the whole porosity range  will be described. Finally, some perspectives are proposed to deal with the multi-dimensional inertial flow by the same asymptotic modeling which supplies nonlinear jump interface conditions. We shall explain why this nonlinear model is likely to yield the global solvability in time.