  We present Giant Metrewave Radio Telescope HI 21cm absorption observations,
of candidate and confirmed damped Lyman-$\alpha$ systems (DLAS). The derived
spin temperatures (T_s) are in all cases $\sim 1000$ K or higher. We have also
collated from the literature a list of DLAS for which 21cm observations exist,
and discuss their implications for the nature of the absorbers.
  A cross-comparison of the 21cm profiles with low ionization metal profiles
shows that 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 DLAS are
considerably higher than those in the Galaxy or nearby spirals. The only
exceptions are DLAS which are known to be associated with the disks of spirals;
these do, in fact, show low spin temperatures. In a multi-phase medium,the
derived T_s is a weighted average of the temperatures of the individual phases.
High derived T_s values are hence to be expected from small, low metallicity
objects, since these objects should have a lower fraction of the cold phase in
their ISM as compared to large galaxies. The high T_s in DLAS is hence
consistent with their observed low metallicities as well as with recent
observations that DLAS are also associated with dwarf/LSB galaxies. Finally, we
suggest that the following trend may be identified: at low redshift, damped
absorption arises from a range of systems, including spiral galaxy disks,
while, at high redshift, absorption occurs predominantly in smaller systems.
(Abridged)
