  We analyse the mass assembly of central galaxies in the EAGLE hydrodynamical
simulations. We build merger trees to connect galaxies to their progenitors at
different redshifts and characterize their assembly histories by focusing on
the time when half of the galaxy stellar mass was assembled into the main
progenitor. We show that galaxies with stellar mass $M_*<10^{10.5}M_{\odot}$
assemble most of their stellar mass through star formation in the main
progenitor (`in-situ' star formation). This can be understood as a consequence
of the steep rise in star formation efficiency with halo mass for these
galaxies. For more massive galaxies, however, an increasing fraction of their
stellar mass is formed outside the main progenitor and subsequently accreted.
Consequently, while for low-mass galaxies the assembly time is close to the
stellar formation time, the stars in high-mass galaxies typically formed long
before half of the present-day stellar mass was assembled into a single object,
giving rise to the observed anti-hierarchical downsizing trend. In a typical
present-day $M_*\geq10^{11}M_{\odot}$ galaxy, around $20\%$ of the stellar mass
has an external origin. This fraction decreases with increasing redshift.
Bearing in mind that mergers only make an important contribution to the stellar
mass growth of massive galaxies, we find that the dominant contribution comes
from mergers with galaxies of mass greater than one tenth of the main
progenitor's mass. The galaxy merger fraction derived from our simulations
agrees with recent observational estimates.
