In this talk we present a new unified approach of general P_N P_M schemes on unstructured meshes in two and three space dimensions for the solution of time-dependent partial differential equations arising in fluid mechanics, such as the compressible Euler and Navier-Stokes equations, the classical and relativistic MHD equations or other PDE systems that govern multifluid and multimaterial flows. The new P_N P_M approach uses piecewise polynomials u_h of degree N to represent the data in each cell. For the computation of fluxes and source terms, another set of piecewise polynomials w_h of degree M>= N is used, which is computed from the underlying polynomials u_h using a reconstruction or recovery operator. The P_N P_M method contains classical high order finite volume schemes (N = 0) and high order discontinuous Galerkin (DG) finite element methods (N = M) just as two particular special cases of a more general class of numerical schemes. Our method also uses a novel high order accurate one-step time discretization, based on a local spacetime discontinuous Galerkin predictor, which is also able to solve PDE with stiff source terms. We show that our method is asymptotic preserving for a linear model system.