The IllustrisTNG simulations (www.tng-project.org) are a novel laboratory to explore galaxy physics and to quantify the assembly and evolution of galaxy populations across an unprecedented range of masses, environments, evolutionary stages and cosmic times. IllustrisTNG is an ongoing program of large cosmological volume simulations where gravity, magnetohydrodynamics, and prescriptions for star formation, stellar evolution, the enrichment of gas with chemical elements beyond Helium, cooling and heating of the gas, galactic outflows and feedback from the supermassive black holes are all taken into account within the LCDM cosmology, i.e. the standard cosmological paradigm. In practice, in these simulations we simultaneously resolve and model the structural details of thousands of galaxies together with the large-scale cosmic web. Here, we summarize our efforts to generate and effectively exploit such simulations, showcase some of the insights they are allowing us to uncover and quantify, and describe our strategies for dissemination.