Disorder is known to have a substantial impact on light transport in optical materials. In particular, when properly tuned, disorder can unveil optical properties that common, periodically patterned materials do not possess. In this paper, a method to realize disordered dielectric materials dubbed Levy glasses, in which light transport is superdiffusive, is presented. The degree of superdiffusion is set by engineering the spatial inhomogeneity of the scatterer density in the material. A model that relates the microscopic parameters to the macroscopic transport properties of Levy glasses is given and the signature of superdiffusion on the transmission profile in a slab configuration is shown experimentally.