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 (z<1) of star formation in the Universe are enabling us to illustrate the important role that cold gas, the fuel for star formation, has played in the assembly of galaxies across cosmic time. Modest sized surveys already provide robust molecular gas detections in hundreds to a few thousand star forming galaxies (SFGs), from redshifts 0-3. Furthermore, spatially resolved spectroscopy in both the (sub)mm and NIR wavelengths now enable us to study the detailed kinematics, star formation, and ISM properties in SFGs on few kpc scales. I will discuss the results from these surveys, and place them in the general context of galaxy formation and evolution.