The propagation of ultrasound through two- and three-dimensional strongly scattering media, with either random or ordered internal structures, has been investigated through experiments and finite element simulations. All media investigated have strong scattering resonances, leading to novel transport behaviour. The two-dimensional samples consist of nylon rods immersed in water. When the nylon rods are arranged in a triangular lattice to form two-dimensional phononic crystals, very unusual dispersion properties are observed when the lattice constant is adjusted so that Bragg and hybridization gaps overlap in frequency. This behaviour is attributed to the competition between two co-existing propagating modes, leading to a new method for tuning bandgap properties and adjusting the transmission by orders of magnitude. The scattering resonance of the nylon rods also leads to unusual Dirac cone properties at the K point of the triangular lattice. The three-dimensional media were fabricated by brazing aluminum beads together to form a disordered porous solid network, with either vacuum or air in the pores, depending on the experiment. This system is of particular interest because it has been shown to exhibit Anderson localization of ultrasound. Two experimental approaches were developed to investigate previously unstudied properties of this system. By directly counting the modes in the frequency domain, the density of states was measured. At intermediate frequencies, the density of states was found to be approximately independent of frequency, while at higher frequencies, the frequency dependence was consistent with traditional density-of-states models. The level statistics of the modes was also investigated to determine the conditions under which level repulsion occurs. By using a laser interferometer to measure the ultrasonic displacements on the surface of a large slab-shaped sample, sub-diffusive behaviour was observed, demonstrating the feasibility of using such measurements to investigate the transition to Anderson localization in these samples.