Over the past decade, increasingly robust estimates of the dense molecular gas content in galaxy populations between redshift z = 0 and the peak of cosmic galaxy/star formation (z ∼ 1-3) have become available. This rapid progress has been possible due to the advent of powerful ground- and space-based telescopes for the 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 as follows: Star-forming galaxies contained much more molecular gas at earlier cosmic epochs than at the present time. The galaxy-integrated depletion timescale 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. 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 cosmological expansion. Another key driver may be global disk fragmentation in high-z, gas-rich galaxies, which ties local free-fall timescales to galactic orbital times and leads to rapid radial matter transport and bulge growth. The low star-formation efficiency inside molecular clouds is plausibly set by supersonic 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. 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.