  We study the properties of black holes and their host galaxies across cosmic
time in the Illustris simulation. Illustris is a large scale cosmological
hydrodynamical simulation which resolves a (106.5 Mpc)^3 volume with more than
12 billion resolution elements and includes state-of-the-art physical models
relevant for galaxy formation. We find that the black hole mass density for
redshifts z = 0 - 5 and the black hole mass function at z = 0 predicted by
Illustris are in very good agreement with the most recent observational
constraints. We show that the bolometric and hard X-ray luminosity functions of
AGN at z = 0 and 1 reproduce observational data very well over the full dynamic
range probed. Unless the bolometric corrections are largely underestimated,
this requires radiative efficiencies to be on average low, epsilon_r <= 0.1,
noting however that in our model radiative efficiencies are degenerate with
black hole feedback efficiencies. Cosmic downsizing of the AGN population is in
broad agreement with the findings from X-ray surveys, but we predict a larger
number density of faint AGN at high redshifts than currently inferred. We also
study black hole -- host galaxy scaling relations as a function of galaxy
morphology, colour and specific star formation rate. We find that black holes
and galaxies co-evolve at the massive end, but for low mass, blue and
star-forming galaxies there is no tight relation with either their central
black hole masses or the nuclear AGN activity.
