Some fifty years after Anderson localization was first proposed, there is currently a resurgence of interest in this phenomenon, which has remained one of the most challenging and fascinating aspects of wave transport in random media. This paper summarizes recent progress in demonstrating the localization of ultrasound in a “mesoglass” made by assembling aluminum beads into a disordered three-dimensional elastic network. In this system, the disorder is sufficiently strong that interference leads to trapping of the waves at intermediate frequencies, as demonstrated by studying three different fundamental aspects of Anderson localization: time-dependent transmission, transverse confinement of the waves, and the statistics of the non-Gaussian intensity fluctuations. Additional ultrasonic experiments have been performed to reveal the multifractal character of the wave functions near the Anderson transition. This is the first time that so many different aspects of localization have been studied simultaneously, providing very convincing evidence for the localization of ultrasonic waves in the presence of disorder in three dimensions, and enabling new aspects of Anderson localization to be studied experimentally.