  We present a new statistical method to determine the relationship between the
stellar masses of galaxies and the masses of their host dark matter haloes over
the entire cosmic history from z~4 to the present. This multi-epoch abundance
matching (MEAM) model self-consistently takes into account that satellite
galaxies first become satellites at times earlier than they are observed. We
employ a redshift-dependent parameterization of the stellar-to-halo mass
relation to populate haloes and subhaloes in the Millennium simulations with
galaxies, requiring that the observed stellar mass functions at several
redshifts be reproduced simultaneously. Using merger trees extracted from the
dark matter simulations in combination with MEAM, we predict the average
assembly histories of galaxies, separating into star formation within the
galaxies (in-situ) and accretion of stars (ex-situ). The peak star formation
efficiency decreases with redshift from 23% at z=0 to 9% at z=4 while the
corresponding halo mass increases from 10^11.8M\odot to 10^12.5M\odot. The star
formation rate of central galaxies peaks at a redshift which depends on halo
mass; for massive haloes this peak is at early cosmic times while for low-mass
galaxies the peak has not been reached yet. In haloes similar to that of the
Milky-Way about half of the central stellar mass is assembled after z=0.7. In
low-mass haloes, the accretion of satellites contributes little to the assembly
of their central galaxies, while in massive haloes more than half of the
central stellar mass is formed ex-situ with significant accretion of satellites
at z<2. We find that our method implies a cosmic star formation history and an
evolution of specific star formation rates which are consistent with those
inferred directly. We present convenient fitting functions for stellar masses,
star formation rates, and accretion rates as functions of halo mass and
redshift.
