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We investigate the coupled formation and evolution of galaxies and their embedded supermassive black holes using state-of-the-art hydrodynamic simulations of cosmological structure formation. For the first time, we self-consistently follow the dark matter dynamics, radiative gas cooling, star formation, as well as black hole growth and associated feedback processes, starting directly from initial conditions appropriate for the LambdaCDM cosmology. Our modeling of the black hole physics is based on an approach we have developed in simulations of isolated galaxy mergers. Here we examine: (i) the predicted global history of black hole mass assembly (ii) the evolution of the local black hole-host mass correlations and (iii) the conditions that allow rapid growth of the first quasars, and the properties of their hosts and descendants today. We find a total black hole mass density in good agreement with observational estimates. The black hole accretion rate density peaks at lower redshift and evolves more strongly at high redshift than the star formation rate density, but the ratio of black hole to stellar mass densities shows only a moderate evolution at low redshifts. We find strong correlations between black hole masses and properties of the stellar systems, agreeing well with the measured local M_BH-sigma and M_BH -M_* relationships, but also suggesting (dependent on the mass range) a weak evolution with redshift in the normalization and the slope. Our simulations also produce massive black holes at high redshift, due to extended periods of exponential growth in regions that collapse early and exhibit strong gas inflows. These first supermassive BH systems however are not necessarily the most massive ones today, since they are often overtaken in growth by quasars that form later. (abridged)