  We study thermal transport in one dimensional spin systems both in the
presence and absence of impurities. In the absence of disorder, all these spin
systems display a temperature dependent Drude peak in the thermal conductivity.
In gapless systems, the low temperature Drude weight is proportional to
temperature and to the central charge which characterizes the conformal field
theory that describes the system at low energies. On the other hand, the low
temperature Drude weight of spin gap systems shows an activated behavior
modulated by a power law. For temperatures higher than the spin gap, one
recovers the linear T behavior akin to gapless systems. For temperatures larger
than the exchange coupling, the Drude weight decays as 1/T^2. We argue that
this behavior is a generic feature of quasi one dimensional spin gap systems
with a relativistic-like low energy dispersion. We also consider the effect of
a magnetic field on the Drude weight with emphasis on the
commensurate-incommensurate transition induced by it. We then study the effect
of nonmagnetic impurities on the thermal conductivity of the dimerized XY chain
and the spin-1/2 two leg ladder. Impurities destroy the Drude peak and the
thermal conductivity exhibits a purely activated behavior at low temperature,
with an activation gap renormalized by disorder. The relevance of these results
for experiments is briefly discussed.
