The universe contains many extremely energetic astrophysical sources that emit particles on timescales from fractions of a second to thousands of years. Studying these sources through gamma-ray and multi-messenger astronomy may help to reveal the answers to several remaining fundamental questions in physics. The High Altitude Water Cherenkov (HAWC) gamma-ray observatory, an extensive air shower array, is well-suited to perform transient searches due to its large field of view and high duty cycle. In this work, a catalog search of all very-high-energy gamma-ray sources in HAWC's field of view is conducted, revealing a total of 65 sources. Then a search for evaporating primordial black holes is performed. None are detected, so upper limits are set on their burst rate density. Finally, the IceCube neutrino observatory is introduced in order to perform a multi-messenger search for flaring gamma rays and neutrinos originating from the same point in the sky. None are detected, so upper limits are placed on the rate density and total energy of sources that produce both gamma rays and neutrinos. Performing these searches contributes to the understanding of the origins of cosmic rays and how astrophysical sources accelerate their particles to the high energies that are observed on Earth.
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