  A technique to determine accurately transport properties of integrable and
non-integrable quantum-spin chains at finite temperatures by Quantum
Monte-Carlo is presented. The reduction of the Drude weight by interactions in
the integrable gapless regime is evaluated. Evidence for the absence of a Drude
weight in the gapless regime of a non-integrable system with longer-ranged
interactions is presented. We estimate the effect of the non-integrability on
the transport properties and compare with recent experiments on one-dimensional
quantum-spin chains.
