  The Fermi-Hubbard model is a key concept in condensed matter physics and
provides crucial insights into electronic and magnetic properties of materials.
Yet, the intricate nature of Fermi systems poses a barrier to answer important
questions concerning d-wave superconductivity and quantum magnetism. Recently,
it has become possible to experimentally realize the Fermi-Hubbard model using
a fermionic quantum gas loaded into an optical lattice. In this atomic approach
to the Fermi-Hubbard model the Hamiltonian is a direct result of the optical
lattice potential created by interfering laser fields and short-ranged
ultracold collisions. It provides a route to simulate the physics of the
Hamiltonian and to address open questions and novel challenges of the
underlying many-body system. This review gives an overview of the current
efforts in understanding and realizing experiments with fermionic atoms in
optical lattices and discusses key experiments in the metallic,
band-insulating, superfluid and Mott-insulating regimes.
