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 k(L) similar to 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 a 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.