Large eddy simulation (LES) seeks to simulate the large structures of a turbulent flow. This is the first monograph which considers LES from a mathematical point of view. It concentrates on LES models for which mathematical and numerical analysis is already available and on related LES models. Most of the available analysis is given in detail, the implementation of the LES models into a finite element code is described, the efficient solution of the discrete systems is discussed and numerical studies with the considered LES models are presented.