Wave transport in strongly scattering, disordered media is investigated experimentally using ultrasonic techniques. Several cases of anomalous wave transport (deviations from conventional diffusion) are studied through experiments on aluminum mesoglass samples, which were designed and created for this purpose. The anomalous wave behaviour is contrasted with conventional diffusion, observed at some frequencies via both transmission and backscattering measurements on wide, thick, slab-shaped samples. The coherent backscattering (CBS) effect is measured experimentally for strongly scattered acoustic waves in three dimensions (3D), and is compared for the first time with theory for diffusive elastic waves in 3D to give an estimate of the diffusion coefficient. At other frequencies, an Anderson localization regime is observed, and is studied in detail. The first experimental study of CBS for localized elastic waves in 3D is presented. By comparing both backscattering and transmission measurements with predictions from a ‘local’ self-consistent theory of localization, the first experimental observation of a complete Anderson mobility gap for elastic waves in 3D is reported. In this mobility gap, large contributions to backscattered intensity from recurrent scattering were observed, enabling the first experimental study of recurrent scattering on its own. The time-dependence of the recurrent scattering, R(t), is shown to agree with theoretical predictions in the diffuse and localized regimes. At the mobility edge, R(t) shows a surprisingly slow decay, prompting further theoretical work. Localization and criticality are also investigated via statistical measurements of ultrasound from cubic mesoglasses of different sizes. Finite-size scaling of multifractal quantities is observed in these cubic samples, and a preliminary fit with theory to determine critical parameters of the Anderson transition is demonstrated. Finally, a sample is which is a candidate to exhibit superdiffusion of ultrasound is studied via a range of experimental techniques, showing subtle deviations from diffusion and opening doors for the next steps in this study.