Hydrodynamical simulations of galaxy formation have now reached sufficient physical fidelity to allow detailed predictions for their formation and evolution over cosmic time. I will review the foundations of this methodology and describe results from a recent effort to construct a new generation of structure formation models, the IllustrisTNG simulations. These numerical simulations have now reached sufficient volume to make accurate predictions for clustering on cosmologically relevant scales, while at the same time being able to compute detailed galaxy morphologies, the enrichment of diffuse gas with metals, or the amplification of magnetic fields during galaxy formation. They also support the notion that supermassive black holes are crucial for understanding the properties of galaxies.