  We have studied numerically the dynamics of a driven elastic interface in a
random medium, focusing on the thermal rounding of the depinning transition and
on the behavior in the $T=0$ pinned phase. Thermal effects are quantitatively
more important than expected from simple dimensional estimates. For sufficient
low temperature the creep velocity at a driving force equal to the $T=0$
depinning force exhibits a power-law dependence on $T$, in agreement with
earlier theoretical and numerical predictions for CDW's. We have also examined
the dynamics in the $T=0$ pinned phase resulting from slowly increasing the
driving force towards threshold. The distribution of avalanche sizes $S_\|$
decays as $S_\|^{-1-\kappa}$, with $\kappa = 0.05\pm 0.05$, in agreement with
recent theoretical predictions.
