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ABSTRACT The initial mass and metallicity of stars both have a strong impact on their fate. Stellar axial rotation also has a strong impact on the structure and evolution of massive stars. In this study, we exploit the large grid of GENEC models, covering initial masses from 9 to 500 $${\rm M}_{\odot }$$ and metallicities ranging from $$Z=10^{-5}$$ (nearly zero) to 0.02 (supersolar), to determine the impact of rotation on their fate across cosmic times. Using the carbon–oxygen core mass and envelope composition as indicators of their fate, we predict stellar remnants, supernova engines, and spectroscopic supernova types for both rotating and non-rotating stars. We derive rates of the different supernova and remnant types considering two initial mass functions to help solve puzzles such as the absence of observed pair-instability supernovae. We find that rotation significantly alters the remnant type and supernova engine, with rotating stars favouring black hole formation at lower initial masses than their non-rotating counterparts. Additionally, we confirm the expected strong metallicity dependence of the fates with a maximum black hole mass predicted to be below 50 $${\rm M}_{\odot }$$ at SMC or higher metallicities. A pair-instability mass gap is predicted between about 90 and 150 $${\rm M}_{\odot }$$, with the most massive black holes below the gap found at the lowest metallicities. Considering the fate of massive single stars has far-reaching consequences across many different fields within astrophysics, and understanding the impact of rotation and metallicity will improve our understanding of how massive stars end their lives, and their impact on the Universe. 

GW190521 challenges our understanding of the late-stage evolution of massive stars and the effects of the pair-instability in particular. We discuss the possibility that stars at low or zero metallicity could retain most of their hydrogen envelope until the pre-supernova stage, avoid the pulsational pair-instability regime and produce a black hole with a mass in the mass gap by fallback. We present a series of new stellar evolution models at zero and low metallicity computed with the Geneva and MESA stellar evolution codes and compare to existing grids of models. Models with a metallicity in the range 0-0.0004 have three properties which favour higher BH masses as compared to higher metallicity models. These are (i) lower mass-loss rates during the post-MS phase, (ii) a more compact star disfavouring binary interaction and (iii) possible H-He shell interactions which lower the CO core mass. We conclude that it is possible that GW190521 may be the merger of black holes produced directly by massive stars from the first stellar generations. Our models indicate BH masses up to 70-75 Msun. Uncertainties related to convective mixing, mass loss, H-He shell interactions and pair-instability pulsations may increase this limit to ~85 Msun.