We present Giant Metrewave Radio Telescope HI 21cm absorption observations, of candidate and confirmed damped Lyman-α systems. The derived spin temperatures T spin are high, in all cases, ∼ 1000 K or higher. We have also collated from the published literature a list of damped absorbers for which 21cm observations exist, and discuss the implications of the observations for the nature of these systems.A cross-comparison of the HI 21cm profiles (which trace the cold gas) with the low ionization metal profiles (which trace all the neutral HI, both cold and warm) shows that in all cases for which both spectra are available, the 21cm absorption coincides in velocity with the deepest metal line feature. This is consistent with models in which the deep metal line features arise from discrete clouds but not with models where the deepest features are the result of velocity crowding.We also find that the typical derived spin temperatures of damped Lyman-α systems are considerably higher than those in the Milky Way or nearby spiral galaxies. The only exceptions are systems which are known to be associated with the disks of spirals; these do, in fact, show low spin temperature. In a multi-phase medium the derived spin temperature is a weighted average of the temperatures of the individual phases. High apparent T spin values are hence to be expected from small, low metallicity objects since these objects should (as per existing theoretical models of the formation of a multi-phase ISM in the Milky Way and high redshift proto-galaxies) have a lower fraction of the cold phase in their ISM as compared to large galaxies. The high T spin is hence consistent with the observed low metallicities of damped Lyman-α absorbers as well as with recent findings that damped absorption is associated with a variety of galaxy types (as opposed to being confined to the disks of large spirals).Finally, although the number of systems for which observations are available is small, we suggest that the following trend may be identified : at low redshift, damped Lyman-α absorption arises from a range of systems, including spiral galaxy disks, while, at high redshift, absorption occurs predominantly in smaller systems.