  The experimental realization of correlated quantum phases with ultracold
gases in optical lattices and their theoretical understanding has witnessed
remarkable progress during the last decade. In this review we introduce basic
concepts and tools to describe the many-body physics of quantum gases in
optical lattices. This includes the derivation of effective lattice
Hamiltonians from first principles and an overview of the emerging quantum
phases. Additionally, state-of-the-art numerical tools to quantitatively treat
bosons or fermions on different lattices are introduced.
