Over the past decade comprehensive and systematic studies of star formation and the gas contents of galaxies during the epochs that are associated with the peak (z~1-3), and subsequent winding down of star formation have enabled us to illustrate the important role that cold gas plays in the assembly of galaxies across cosmic time. These studies show that star forming galaxies contained significantly more molecular gas at earlier cosmic epochs than at the present time. Global rates of galaxy gas accretion, which vary with cosmological expansion, primarily drive this increase in cold gas and star formation rates in the dominant main sequence galaxy population. Studies also show that the molecular gas depletion time depends mainly on redshift or Hubble time, and at a given z, on the vertical location of a galaxy relative to the “star formation main sequence”. In this talk, I will discuss various strategies and methods used to determine the evolution of cold gas contents, and discuss the latest gas scaling relations with redshift, star formation and stellar mass. I will also discuss how simple gas regulator models successfully predict the combined evolution of molecular gas fractions, star formation rates, galactic winds, and gas phase metallicities.