40 years ago, Richard Feynman outlined his vision of a quantum simulator for carrying out complex calculations of physical problems. Today, his dream has become a reality and a highly active field of research across different platforms ranging from ultracold atoms and ions, to superconducting qubits and photons. In my lecture, I will outline how ultracold atoms in optical lattices started this vibrant and interdisciplinary research field 20 years ago and now allow probing quantum phases in- and out-of-equilibrium with fundamentally new tools and single particle resolution. Novel (hidden) order parameters, entanglement properties, full counting statistics or topological features can now be measured routinely and provide deep new insight into the world of correlated quantum matter. I will introduce the measurement and control techniques and a few recent applications regarding quantum simulations of condensed matter systems, namely the Fermi Hubbard model, which plays an essential role in the context of High-Tc superconductivity, experiments on new dynamical phases of matter, as well as topological systems.