  After reviewing the main features of anomalous energy transport in 1D
systems, we report simulations performed with chains of noisy anharmonic
oscillators. The stochastic terms are added in such a way to conserve total
energy and momentum, thus keeping the basic hydrodynamic features of these
models. The addition of this "conservative noise" allows to obtain a more
efficient estimate of the power-law divergence of heat conductivity kappa(L) ~
L^alpha in the limit of small noise and large system size L. By comparing the
numerical results with rigorous predictions obtained for the harmonic chain, we
show how finite--size and --time effects can be effectively controlled. For low
noise amplitudes, the alpha values are close to 1/3 for asymmetric potentials
and to 0.4 for symmetric ones. These results support the previously conjectured
two-universality-classes scenario.
