Catenary functions play pivotal roles in describing the electromagnetic vectors, intensity distribution, and dispersion of structured light on the subwavelength scale. In this article, the history, basic theories, functional devices, and applications of catenary functions in optics and electromagnetics are reviewed. First, the catenary fields generated by the coupling of evanescent waves are discussed, together with their applications in flat optics, super-resolution imaging, and lithography. Second, the relation between catenary optical fields and catenary dispersion is interpreted using static approximation, which provides a general guideline for realizing the quick design of functional metasurfaces, such as broadband electromagnetic absorbers and flat antennas. Third, the streamlines of vectorial optical fields are described using logarithmic cosine catenary functions. Catenary shaped quasicontinuous structures, which can be used to form functional metasurfaces, and their applications in spin optics are reviewed. Finally, the challenges and promising research directions of catenary optics and catenary electromagnetics are outlined. © 2020 Wiley-VCH GmbH