It has long been recognized that post-main-sequence stellar evolution and stellar death have dramatic effects on a planetary system, whether by swallowing and incinerating nearby planets or by provoking dynamical reorganization of orbital architectures. In recent years, direct observations of planetary systems around post-MS stars have demonstrated these processes in action. In this dissertation, I study several aspects of post-MS planetary evolution, using simplified physical models to address various puzzles posed by these observations and provide predictions for recent and forthcoming observations. Numerous topics within theoretical astrophysics intersect in this work, including celestial mechanics, fluid dynamics, planet formation, radiative processes, and stellar structure and evolution. Chapters 2, 3, and 4 pertain to phenomenon of white dwarf (WD) pollution, which provides indirect evidence for the existence of planets around the remnants of Sun-like and intermediate-mass stars. In Chapter 2, I study the orbital circularization of a planetesimal passing close to a WD on an eccentric orbit, in order to understand the history of two polluted WD systems containing surviving asteroids on short-period orbits. In Chapter 3, I consider a dynamical process known as secular chaos that may operate in WD systems, in order to explain the longstanding puzzle of how WD pollution is sustained on timescales of several Gyr. In Chapter 4, I examine the evolution of extrasolar Oort clouds (large-scale comet reservoirs) around WDs, to account for the curious absence of volatile-enriched polluted WDs. Chapters 5 and 6, meanwhile, are dedicated to the extraordinary object WD1856+534b, a Jupiter-sized planet orbiting a WD with a period of just 1.4 days, and the effort to understand its origin on both theoretical and observational bases. In Chapter 5, I demonstrate that the object could have undergone high-eccentricity tidal migration via the Lidov–Kozai effect, due to gravitational interactions with its known stellar companions. In Chapter 6, I explore an alternative scenario in which the object was engulfed by its red-giant host star and survived by facilitating the expulsion of the stellar envelope, a process known as common-envelope evolution, using the Modules for Experiments in Stellar Astrophysics software instrument. Because planet engulfment is a natural consequence of stellar evolution, in this chapter I also study the potential observational signatures of ongoing engulfment events in the Galaxy. In Chapter 7, I summarize my results and briefly describe ongoing and future work.