Transient sources of very-high-energy gamma rays are short-lived astrophysical phenomena often associated with catastrophic events that change their brightness over relatively short timescales. The search for and study of such objects, especially in the TeV energy regime, has the possibility to shed light not only on the physics at play within the enigmatic, chaotic environments that produce such emission, but also to answer several remaining questions in fundamental physics. In this dissertation, we leverage the sensitivity and characteristics of the High-Altitude Water Cherenkov (HAWC) Observatory in pursuit of gaining insight into these areas.

The HAWC Observatory, located on the side of the side of the Sierra Negra volcano in Puebla, Mexico at an altitude of 4,100 m above sea level, is an extensive-air-shower array sensitive to gamma rays from ~0.1 to >100 TeV that has been in operation since March of 2015. It has a wide field of view of ~2 sr at any one time and, combined with its large operational duty cycle (>95%), observes 2/3 of the sky every day. HAWC operates using the water-Cherenkov detection technique with 1,200 photomultiplier tubes (PMTs) in two different sizes to detect Cherenkov emission from secondary air-shower particles. Herein, we present an improved characterization for the larger of these two PMT models for inclusion within the Monte Carlo simulation of the HAWC Observatory, as well as the custom testing apparatus designed and constructed for this purpose.

With HAWC's wide field of view, near-continuous uptime, and large archival dataset, it serves as an ideal observatory with which to search for transient sources of all kinds. We apply these advantages to perform searches for two types of transient sources--- Primordial Black Holes (PBHs) and Gamma-Ray Bursts (GRBs). The first of these, a search for emission signatures of evaporating PBHs, is performed on 959 days of HAWC data for remaining lifetimes of 0.2, 1, 10, and 100 s, assuming radiation development according to the Standard Emission Model. We show that previous attempts to perform searches for transient searches similar to PBHs with HAWC were oversampling at detrimental levels and improve upon that method to achieve greater statistical rigor. Finding no significant emission for any duration, we place upper limits at the 99% confidence level on the local burst rate density.

For the second of these source types, we apply the low-energy improvements recently made to the HAWC data reconstruction procedure to search for very-high-energy emission within the first 0.1, 1, 10, and 100 s of emission for 93 GRBs within HAWC's field of view at their reported T₀ over the first 7 years of HAWC operations. This search is performed using permutations of four different assumed redshift values and four different assumed spectral indices. Finding no significant emission for any duration under any set of assumption parameters, we place upper limits at the 95% confidence level on the intrinsic flux for all GRBs. For those GRBs with external flux models available from other gamma-ray detectors, we compare the HAWC limits to those models in order to constrain the possible emission in the TeV regime with respect to that at lower energy values. We also perform a follow-up execution of this analysis with start times shifted to match external model start times which differed from T₀. Again finding no significant emission, we place upper limits at the 95% confidence level on the intrinsic flux for all parameter sets and for all external start times for those GRBs HAWC was most likely to have seen.

Finally, we speculate about the future of searches for PBHs and GRBs with the next-generation wide-field-of-view instrument, the Southern Wide-field Gamma-ray Observatory (SWGO), presenting projected performance for these two types of transient sources.