Leaky waves have been among the most active areas of research in microwave engineering over the second half of the 20th century. They have been shown to dominate the near-field of several open wave-guiding structures, of great interest to tailor their radiation, guidance and filtering properties. The elegant theoretical analyses and deep physical insights in this area, developed in an era in which computational resources were limited, represent a fundamental scientific legacy that is still extremely relevant in today's engineering society and beyond. In this regard, the relevance of leaky-wave concepts has been increasingly recognized in recent times over a broader scientific community, including optics and physics societies. In this paper, after revisiting the fundamental concepts of leaky-wave theory, we discuss and connect different relevant research activities in which leaky-wave concepts have been applied, with the goal of facilitating multidisciplinary interactions on these topics. In addition to the canonical microwave applications of leaky waves, particular attention is devoted to a few areas of interest in modern optics, such as directive optical antennas, extraordinary optical transmission, and embedded scattering eigenvalues, in which leaky waves play a fundamental role.