Recent experimental advances in controlling quantum coherence present new opportunities to realize quantum simulation of complex quantum many-body systems. Quantum simulators based on a variety of architectures, including ultracold atoms and molecules in optical lattices, trapped ions, Rydberg gases, isolated spins in solids, and superconducting circuits, proffer rich playgrounds for Hamiltonian engineering and time-dependent control of many-body quantum systems. Many of these systems can routinely achieve parameter regimes and sizes that are seemingly intractable to simulate on a classical computer.