The purpose of the presentation is to review Residual Distribution Schemes on simplex elements, as a technology for solving hyperbolic conservation laws. The focus will be on the open problems and bottlenecks that require further research. First, a historical sketch will be given of the different steps that have led to the present framework, starting from landmark N-scheme (Roe 1986): scalar schemes (N-, LDA and nonlinear schemes), extension to systems through wave modeling and through matrix schemes, conservative Roe-linearization for the Euler equations, interpretation as a class of Petrov-Galerkin Finite Element methods. Then, more recent issues like will be reviewed like: conservation for general conservation systems, consistent upwind treatment of source terms unsteady time-space schemes. Experience gathered in solving compressible and incompressible Euler and Navier-Stokes, ideal MHD equations and two-fluid two-phase flow models will be discussed. Finally, the issues which in our view remain open or unsatisfactory are outlined:
- the difficult compromise between conservation properties and monotonicity
- discrete entropy satisfaction and entropy fix, carbuncle phenomenon
- development of faster relaxation methods
- schemes on quadrilateral/hexahedral elements
- higher order schemes (higher than 2)
- unsteady problems.