  We explore the host galaxies of compact-object binaries (black hole--black
hole binaries, BHBs; neutron star--black hole binaries, NSBHs; double-neutron
stars; DNSs) across cosmic time, by means of population-synthesis simulations
combined with the Illustris cosmological simulation. At high redshift
($z\gtrsim{}4$) the host galaxies of BHBs, NSBHs and DNSs are very similar and
are predominantly low-mass galaxies (stellar mass $M<10^{11}$ M$_\odot$). If
$z\gtrsim{}4$ most compact objects form and merge in the same galaxy, with a
short delay time. At low redshift ($z\leq{}2$), the host galaxy populations of
DNSs differ significantly from the host galaxies of both BHBs and NSBHs. DNSs
merging at low redshift tend to form and merge in the same galaxy, with
relatively short delay time. The stellar mass of DNS hosts peaks around
$\sim{}10^{10}-10^{11}$ M$_\odot$. In contrast, BHBs and NSBHs merging at low
redshift tend to form in rather small galaxies at high redshift and then to
merge in larger galaxies with long delay times. This difference between DNSs
and black hole binaries is a consequence of their profoundly different
metallicity dependence.
