The Anderson localization literature in structural acoustics has to date been concerned largely with applications to the vibrations of one dimensional structures, whether mono-coupled or multi-coupled, and to steady state responses in such systems. This paper presents a brief tutorial on the theory of wave localization in one and higher dimensions with an emphasis on the scaling theory of localization. It then reviews the acoustic and optical literature on wave localization with an emphasis on diffuse time domain responses to transient loads. Numerical and laboratory experiments demonstrating localization in higher dimensions and investigating the time-domain behavior of such systems are discussed. Scaling theory is shown to provide predictions for localization lengths in weakly disordered multi-coupled systems, and for localization lengths in weakly disordered two-dimensional systems as well. Theoretical arguments for rates of diffuse transport are contrasted with the experimental evidence. The paper concludes with a discussion of wave energy confinement in non-localizing disordered systems.