Gas-rich galaxy centers like our own Milky Way's Central Molecular Zone (CMZ) are amongst the most extreme, yet poorly understood, environments for star formation. These regions are affected by large-scale in- and out-flows of gas, by nuclear starburst and/or active galactic nuclei, which makes them unique laboratories in the local universe to study star formation at the highest gas and stellar mass densities, intense radiation fields, and short (<50 Myr) dynamical timescales. CMZs are compact (<1kpc) and subject to high extinction, which requires near- to mid-infrared observations to study their individual star-forming sites. The lack of data that resolves the molecular gas reservoirs and star formation sites has prevented establishing a coherent picture of this star formation regime – a situation that has now radically changed by JWST. At the same time, ALMA has both the sensitivity and spatial resolution to map the underlying gas distribution. This ERC project will develop a first synthetic view of extragalactic CMZs (eCMZs) by zooming in to centers of 37 nearby galaxies from a sample that has been globally well characterized through the PI's PHANGS survey. The PI has assembled in a set of observational campaigns an absolutely unique ALMA+HST+JWST dataset, with unprecedented 5-30pc resolution. Rigorous analysis of these data will provide the urgently needed observational framework to distinguish between theories for star formation under extreme CMZ conditions: it may either argue for a highly time-variable star formation dominated by local processes, or for a more steady-state evolution regulated by the larger-scale environment.