  We use the Illustris Simulations to gain insight into the build-up of the
outer, low-surface brightness regions which surround galaxies. We characterize
the stellar haloes by means of the logarithmic slope of the
spherically-averaged stellar density profiles, alphaSTARS at z=0, and we relate
these slopes to the properties of the underlying Dark-Matter (DM) haloes, their
central galaxies, and their assembly histories. We analyze a sample of ~5,000
galaxies resolved with more than 5x10^4 particles each, and spanning a variety
of morphologies and halo masses (3x10^11 < Mvir < 10^14 Msun). We find a strong
trend between stellar halo slope and total halo mass, where more massive
objects have shallower stellar haloes than the less massive ones (-5.5 \pm 0.5
< alphaSTARS <-3.5 \pm 0.2 in the studied mass range). At fixed halo mass, we
show that disk-like, blue, young, and more massive galaxies are surrounded by
significantly steeper stellar haloes than elliptical, red, older, and less
massive galaxies. Overall, the stellar density profiles fall off much more
steeply than the underlying DM, and no clear trend holds between stellar slope
and DM halo concentration. However, DM haloes which formed more recently, or
which accreted larger fractions of stellar mass from infalling satellites,
exhibit shallower stellar haloes than their older analogs with similar masses,
by up to Delta(alphaSTARS) ~ 0.5-0.7. Our findings, combined with the most
recent measurements of the strikingly different stellar power-law indexes for
M31 and the Milky Way, appear to favour a massive M31, and a Milky Way
characterized by a much quieter accretion history over the past 10 Gyrs than
its companion.
