We present a new anatomically-constrained local face model and fitting approach for tracking 3D faces from 2D motion data in very high quality. In contrast to traditional global face models, often built from a large set of blendshapes, we propose a local deformation model composed of many small subspaces spatially distributed over the face. Our local model offers far more flexibility and expressiveness than global blendshape models, even with a much smaller model size. This flexibility would typically come at the cost of reduced robustness, in particular during the under-constrained task of monocular reconstruction. However, a key contribution of this work is that we consider the face anatomy and introduce subspace skin thickness constraints into our model, which constrain the face to only valid expressions and helps counteract depth ambiguities in monocular tracking. Given our new model, we present a novel fitting optimization that allows 3D facial performance reconstruction from a single view at extremely high quality, far beyond previous fitting approaches. Our model is flexible, and can be applied also when only sparse motion data is available, for example with marker-based motion capture or even face posing from artistic sketches. Furthermore, by incorporating anatomical constraints we can automatically estimate the rigid motion of the skull, obtaining a rigid stabilization of the performance for free. We demonstrate our model and single-view fitting method on a number of examples, including, for the first time, extreme local skin deformation caused by external forces such as wind, captured from a single high-speed camera.