Recent experiments on Lévy glasses showed that a general theory for random walks in disordered media with quenched Lévy distributed spacings is still missing. Here we analyze the 1D case (scatterers distributed along a line with power-law distribution of the gaps between them) and we are able to obtain the asymptotic behavior of the mean square displacement as a function of the exponent characterizing the distribution of the scatterers, using the scaling relations for the random walk probability and for the resistivity in the equivalent electric problem. We also check these results against numerical simulations, with excellent agreement.