Most finite element or finite volume software is built around a fixed mesh data structure. Therefore, each software package can only be used efficiently for a relatively narrow class of applications. For example, implementations supporting unstructured meshes allow the approximation of complex geometries but are in general much slower and require more memory than implementations using structured meshes. In this paper we show how a generic mesh interface can be defined such that one algorithm, e. g. a finite element discretization scheme, can work efficiently on different mesh implementations. These ideas have also been extended to vectors and sparse matrices where iterative solvers can be written in a generic way using the interface. These components are available within the “Distributed Unified Numerics Environment” (DUNE).