Quantum systems with many strongly correlated degrees of freedom are fundamentally different from systems with only a few particles. This is evident in many condensed matter systems, where the interplay of many degrees of freedom leads to the emergence of theoretically interesting and practically useful quantum phases such as superconductivity, charge order, and magnetism. Because of the absence of small parameters and the large number of degrees of freedom, understanding these phases with traditional tools of many-body theory has proven to be challenging. This talk will show how numerical methods can be used to simulate strongly correlated quantum systems and explore the connections to simple analytical theories and experiment. In particular, we will show quantitative simulations for lattice model systems and their comparison to ultracold atomic gas experiment, and an analysis of the interplay between superconductivity and the so-called pseudogap phase in high-temperature superconducting materials.