Star-formation and galaxy growth are complex phenomenon regulated by accretion of cold gas from the cosmic web, mergers big and small, star-formation in dense molecular clouds, and feedback - both from star-formation and super- massive blackholes (SMBHs). In this thesis, we probe different aspects of star- formation, and feedback from Active Galactic Nuclei (AGN), using observations of the cold molecular gas, primarily traced by CO and other dense molecular gas tracers. The research presented here takes place in two arcs. In the first, we probe the galaxy-wide molecular gas reservoirs at high-redshift (z ∼ 1 − 3). Galaxies at those redshifts were forming stars much more vigorously than at z ~ 0, and we use observations of the dense molecular gas, traced by HCN, to test whether the efficiency of star-formation is also correspondingly higher. We find that the increase in star-formation is driven in large part by the extended gas reservoirs, with little enhancement in the efficiency of star-formation. In the second part of the thesis, we explore the molecular gas content of nearby (z ~ 0.15) massive, IR-luminous galaxies, probing both their galaxy-wide content as well as the gas in the central ~ 50 pc. Such dust-obscured systems boast extraordinarily compact, obscured nuclei, which produce the bulk of their IR emission. The nuclear star-formation and accretion onto the SMBH in the nuclei give rise to massive molecular outflows, which can regulate both star- formation and SMBH growth on short timescales (~ 10 Myr) and can give rise to the observed correlation between stellar bulge and SMBH properties. Our observations reveal some of the earliest stages of this feedback, and that the most extreme of such outflows are powered by both AGN and starburst activity.