2023 / The L and T spectral types extend the traditional Harvard stellar classification system to include objects that are less massive, cooler, and spectroscopically distinct from the latest M dwarfs. L and T dwarfs span the transition from the stellar regime to that of brown dwarfs and planets and were a missing component of the stellar classification sequence until the proliferation of near-infrared (NIR) array detectors, which probe the wavelengths where L and T dwarfs give off the majority of their flux (∼1−2.5 μm). For the same reason, L and T dwarfs remain a mostly unexplored parameter space for exoplanet surveys, which have traditionally been performed at optical wavelengths. Predictions differ as to whether or not L and T dwarfs host a significant population of short-period super-Earth and mini-Neptune planets, a tension that can only be resolved by searching for satellites around L and T dwarfs. In this dissertation, I report the results of just such a survey: the Perkins INfrared Exosatellite Survey (PINES). PINES is a NIR search for transiting satellites around L and T dwarfs and it represents the largest photometric monitoring campaign of L and T dwarfs, to date. I describe the design of the survey, its implementation on Boston University's 1.83-m Perkins Telescope Observatory, and its ongoing operation, which has yielded light curves of over 130 L and T dwarfs. I present a systematic search for transit events in these light curves. I found one plausible candidate, which, if confirmed, would represent the first transiting planet discovered around a brown dwarf and would point to enhanced planet occurrence rates for L and T dwarfs compared to earlier-type stars. Finally, I investigated the potential for detecting transiting planets around L and T dwarfs with the upcoming Nancy Grace Roman Space Telescope, which will perform a large-scale time domain survey of the Galactic bulge at NIR wavelengths. I show that this survey will detect few transiting planets around L and T dwarfs (if any) and that ground-based surveys like PINES will remain the best options for discovering such planets in the coming decade.