  Over the past decade increasingly robust estimates of the dense molecular gas
content in galaxy populations between redshift 0 and the peak of cosmic
galaxy/star formation from redshift 1-3 have become available. This rapid
progress has been possible due to the advent of powerful ground-based, and
space telescopes for combined study of several millimeter to far-IR, line or
continuum tracers of the molecular gas and dust components. The main
conclusions of this review are:
  1. Star forming galaxies contained much more molecular gas at earlier cosmic
epochs than at the present time.
  2. The galaxy integrated depletion time scale for converting the gas into
stars depends primarily on z or Hubble time, and at a given z, on the vertical
location of a galaxy along the star-formation rate versus stellar mass
"main-sequence" (MS) correlation.
  3. Global rates of galaxy gas accretion primarily control the evolution of
the cold molecular gas content and star formation rates of the dominant MS
galaxy population, which in turn vary with the cosmological expansion. A second
key driver may be global disk fragmentation in high-z, gas rich galaxies, which
ties local free-fall time scales to galactic orbital times, and leads to rapid
radial matter transport and bulge growth. Third, the low star formation
efficiency inside molecular clouds is plausibly set by super-sonic streaming
motions, and internal turbulence, which in turn may be driven by conversion of
gravitational energy at high-z, and/or by local feedback from massive stars at
low-z.
  4. A simple 'gas regulator' model is remarkably successful in predicting the
combined evolution of molecular gas fractions, star formation rates, galactic
winds, and gas phase metallicities.
