Adaptive anisotropic refinement of finite element meshes allows to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem. We present a new approach to adaptive refinement and demonstrate that this allows to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.:1 Introduction 2 Extension of FEM ansatz spaces 3 Optimality of the extension 4 Application 1: graded refinement 5 Application 2: anisotropic refinement in 2D 6 Numerical experiments 7 Conclusions and outlook