  One of the last missing pieces in the puzzle of galaxy formation and
evolution through cosmic history is a detailed picture of the role of the cold
gas supply in the star-formation process. Cold gas is the fuel for star
formation, and thus regulates the buildup of stellar mass, both through the
amount of material present through a galaxy's gas mass fraction, and through
the efficiency at which it is converted to stars. Over the last decade,
important progress has been made in understanding the relative importance of
these two factors along with the role of feedback, and the first measurements
of the volume density of cold gas out to redshift 4, (the "cold gas history of
the Universe") has been obtained. To match the precision of measurements of the
star formation and black-hole accretion histories over the coming decades, a
two orders of magnitude improvement in molecular line survey speeds is required
compared to what is possible with current facilities. Possible pathways towards
such large gains include significant upgrades to current facilities like ALMA
by 2030 (and beyond), and eventually the construction of a new generation of
radio-to-millimeter wavelength facilities, such as the next generation Very
Large Array (ngVLA) concept.
