As the nascent fields of fast radio burst and gravitational wave astronomy grow in observational relevance, the lensing of these compact, coherent sources of radiation will play an important role in shaping our understanding of some of the fundamental questions of cosmology. Questions such as: what is the large-scale behaviour of space-time? what is the small-scale nature of dark matter? and what is the physical state of the invisible baryonic component of the universe? As such, the study of wave optics in astrophysical lensing contexts has recently received revived interest, as these sources are expected to exhibit wave effects when they are lensed in a wide variety scenarios. I will discuss work that further develops the theory of coherent wave lensing by expanding upon the mathematical framework of Picard-Lefschetz theory for the evaluation of oscillatory path integrals. This novel technique reveals the fundamental importance of complex images in governing wave optics and the transition from diffractive to refractive optics. In addition to developing new theoretical tools, I will discuss new applications of wave lensing from the detection of exo-galactic exoplanets, to extreme precision measurements of cosmic expansion, to probing the geometric structure of plasma in the interstellar and circumgalactic medium.