"Ultracold atoms in optical lattices offer previously unparalleled possibilities to simulate quantum many-body effects with almost full control and detection at the single particle level. Recently, in-situ-imaging with single-atom resolution has become available for bosonic rubidium atoms, but an experimental demonstration of the single-site-resolved detection of fermions is still missing. The aim of this research project is to detect and to manipulate strongly correlated fermionic quantum systems in an optical lattice by means of a specially designed microscope objective with single-site and single-atom resolution. This will allow us to simulate several seminal models of condensed matter physics, in particular the Fermi-Hubbard model, which is conjectured to be a key model for high-Tc superconductivity. Probing strongly correlated systems at this most fundamental, single-particle level is a fantastic advantage over the present techniques which use time-of-flight images. Gaining access to the in-trap atom distribution of the fermions with single-atom resolution will enable the precise characterization of spatial order, temperature, and entropy distribution of fermionic many-body states, such as fermionic Mott insulators, band insulators, metallic phases or Néel antiferromagnets. Furthermore, spin manipulations on the scale of individual lattice sites can locally perturb the system, and the ensuing dynamical in-trap evolution can be recorded with unprecedented resolution. By simulating condensed matter systems with ultracold fermionic atoms in optical lattices, this project will help to make Richard Feynman’s vision of a quantum simulator become a reality."