Deep optical and near-IR surveys have traced the star formation history of the Universe as a function of environment, stellar mass, and galaxy activity (AGN and star formation), back to cosmic reionization and the first galaxies (z ∼ 6 to 8). While progress has been truly impressive, optical and near-IR studies of primeval galaxies are fundamentally limited in two ways: (i) obscuration by dust can be substantial for rest-frame UV emission, and (ii) near-IR studies reveal only the stars and ionized gas, thereby missing the evolution of the cool gas in galaxies, the fuel for star formation. Line and continuum studies at centimeter through submillimeter wavelengths address both these issues, by probing deep into the earliest, most active and dust obscured phases of galaxy formation, and by revealing the molecular and cool atomic gas. We summarize the techniques of radio astronomy to perform these studies, then review the progress on radio studies of galaxy formation. The dominant work over the last decade has focused on massive, luminous starburst galaxies (submm galaxies and AGN host galaxies). The far infrared luminosities are ∼ 10 13 L ⊙ , implying star formation rates, SFR ≥ 10 3 M ⊙ year −1 . Molecular gas reservoirs are found with masses: M(H 2 ) > 10 10 (α/0.8) M ⊙ . The CO excitation in these luminous systems is much higher than in low redshift spiral galaxies. Imaging of the gas distribution and dynamics suggests strongly interacting and merging galaxies, indicating gravitationally induced, short duration (≤ 10 7 year) starbursts. These systems correspond to a major star formation episode in massive ⋆ The Very Large Array of the National Radio Astronomy Observatory, is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc -2 -galaxies in proto-clusters at intermediate to high redshift. Recently, radio observations have probed the more typical star forming galaxy population (SFR ∼ 10 2 M ⊙ year −1 ), during the peak epoch of Universal star formation (z ∼ 1.5 to 2.5). These observations reveal massive gas reservoirs without hyper-starbursts, and show that active star formation occurs over a wide range in galaxy stellar mass. The conditions in this gas are comparable to those found in the Milky Way disk. A key result is that the peak epoch of star formation in the Universe also corresponds to an epoch when the baryon content of star forming galaxies was dominated by molecular gas, not stars. We consider the possibility of tracing out the dense gas history of the Universe, and perform initial, admittedly gross, calculations. We conclude with a description and status report of the Atacama Large Millimeter Array, and the Expanded Very Large Array. These telescopes represent an order of magnitude, or more, improvement over existing observational capabilities from 1 GHz to 1 THz, promising to revolutionize our understanding of galaxy formation.