The gas and dynamical masses of damped Lyman-alpha absorbers (DLAs) have been open questions in the field of galaxy evolution for more than three decades. This talk will describe new results from Arecibo and ALMA searches for HI 21cm, CO and CII-158 micron searches emission from a sample of DLAs at, respectively, low (z < 0.1), intermediate (z~0.7) and high (z~4) redshifts. Our HI 21cm observations of the DLAs at z<0.1 yield normal gas masses, <~ 5 x 10^9 solar masses, but very high gas-to-stars mass ratios, ~ 5-100, far higher than in normal galaxies. For the absorbers at intermediate redshifts, we obtain large molecular gas masses in the six systems with CO detections, despite their low optical star formation rates. We also obtain high star formation rates (based on dust continuum emission) and high CII-158 micron line luminosities for the DLA host galaxies at z~4. For the CO and CII-158 micron detections, the impact parameters of the host galaxies are high, 15-45 kpc, far larger than expected based on earlier studies. These are the first CO and CII-158 micron detections in DLA hosts, providing a new window on physical conditions in the absorbers and yielding a new tool to identify DLA host galaxies at high redshifts.