The puzzling and important phenonenon of wave localization arises in many physical and mathematical contexts, with applications range from the quantum mechanics of electrical conduction through the design of optical devices to the construction of noise abatement systems, to name but a few. Although studied by physicists and mathematicians for the better part of a century, localization of eigenmodes is still not fully understood nor controlled. In this talk we will describe recent major strides which have been made towards a comprehensive theory. In particular, it is now possible to predict and control the spectrum--both the eigenfunctions and the eigenvalues--of a large class of elliptic PDE, such as Schrodinger operators with random potentials. The talk will feature numerous high fidelity large scale finite element computations which have played a crucial role in guiding our understanding, validating theoretical results, and highlighting mysteries as yet unexplained.​