Gas and star formation in galaxies are intimately linked to one another. Molecular hydrogen gas is the material out of which stars form, while the process of forming stars, in turn, depletes the reservoirs of gas in galaxies and builds up their stellar mass. Observations of star formation in galaxies over time indicate that they must form stars for timescales longer than would be expected from their gas content and star formation rates, indicating that processes that replenish the star forming fuel must be present. The focus of this thesis is on two components of this qualitative picture: the molecular hydrogen gas content of galaxies over time, and the link between gas and star formation in galaxies resembling those observed at the epoch of most active star formation.

First, I present a systematic search for serendipitous carbon monoxide emitting sources in the second Plateau de Bure High-z Blue-Sequence Survey (PHIBSS2). These observations presented an opportunity to quantify the mass density of molecular gas in galaxies as a function of time, and to link this to the star formation history of the Universe. I use a match-filter technique to systematically detect 67 serendipitous sources, after which I characterize their properties, creating a catalog of their redshifts, line widths, fluxes, estimations of the detection reliability, and completeness of the detection algorithm. I find that these serendipitous sources are unrelated to the primary sources that were targeted by PHIBSS2, and use the catalog to construct luminosity functions spanning a redshift range from $\sim 0.3-5$. From these luminosity functions, I place constraints on the molecular hydrogen content in galaxies over cosmic time. My work presents one of the first attempts to use existing observations for this measurement and yields results that are consistent with previous studies, while demonstrating the scientific power of large, targeted surveys.

Next, I study a sample of rare, nearby galaxies that are most similar to those we observe at the peak of cosmic star forming activity that occured $\sim 10$ billion years ago. These galaxies are drawn from the DYnamics of Newly Assembled Massive Objects (DYNAMO) survey, and their proximity to us allows for very detailed studies of their massive star forming clumps. I use observations from the Hubble Space Telescope (HST) to measure colors that are sensitive to stellar population age and extinction. From these measurements, I find that clumps in DYNAMO galaxies have colors that are most consistent with very young centers and outskirts that appear systematically older, by as much as 150~Myr in some cases. I attribute this age difference to the presence of ongoing star formation in the centers of clumps that maintains the population of massive, short-lived stars and gives rise to colors consistent with young ages. Furthermore, I find that within the disks of their host galaxies, younger clumps are preferentially located far from galaxy centers, while older clumps are preferentially located closer to the centers. These results are consistent with hydrodynamic simulations of high-redshift clumpy galaxies that predict clumps form in the outskirts of galaxies via a violent disk instability, and as they age, migrate to the centers of galaxies where they merge and contribute to the growth of galactic bulges.

Building on this study, I combine observations of DYNAMO galaxies from the HST and the Atacama Large Millimeter/sub-millimeter Array (ALMA) to trace molecular hydrogen gas and star formation. I link these observations to measurements of the molecular gas velocity dispersions to test theories of star formation. I find that compared to local samples of ``normal'' star forming galaxies, DYNAMO systems have consistently high velocity dispersions, molecular gas surface densities, and star formation rate surface densities. Indeed, throughout their disks, DYNAMO galaxies are comparable to the centers of local star forming galaxies. Stellar bar driven gas flows into the centers of galaxies in these local samples may give rise to the high observed velocity dispersions, and gas and star formation rate surface densities. For DYNAMO galaxies, the widespread elevated values of these parameters may be driven by galactic-scale gas inflows, which is predicted by theories. Finally, current theories of star formation, such as the feedback regulated model, assume that turbulence dissipates on timescales proportional to the angular velocity of a galaxy (eddy or crossing time). Yet, I find such models have difficulty reproducing the DYNAMO measurements, and thus conclude that the turbulent dissipation timescale in DYNAMO galaxies must scale with galactocentric radius.