Organizers: Herman Deconinck and Mario Ricchiuto (VKI). The course focuses on very high-order numerical schemes for simulating fluid flow and convection dominated physics in general, in particular those methods allowing the use of unstructured grids. The course offers a wide overview and recent developments of different methods with the ultimate objective to create a bridge between applied mathematics (numerical analysis) and fluid dynamics (real application). Introductory courses present Finite Volume and Finite Element Methods in a unified framework, and discuss discretizations that exploit the structure of the governing equations. Then a large number of (unstructure grid) frameworks for constructing schemes of very high-order of accuracy are introduced, and their state of the art discussed: this includes: Discontinuous Galerkin and ADER Discontinuous Galerkin schemes, Spectral Element and Spectral Volume methods, Compact Finite Difference schemes, Residual Based Compact and Residual Distribution schemes and finally well-balanced WENO discretizations.