The puzzling phenomenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating media. Localization arises in many physical and mathematical systems and has many important implications and applications. It is closely associated with Philip Anderson (1923-2020) who received the Nobel prize in 1977 for his 1958 discovery of localization of the eigenfunctions of the Schrödinger equation with a random potential, a process which dramatically affects the electrical properties of materials disordered by impurities. Despite this long history, many aspects of localization remain mysterious even today. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential. We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.