Thermonuclear burning of hydrogen on white dwarfs (WDs) is an inevitable occurence for accreting WDs in binary systems. After the onset of thermally-unstable nuclear burning, the WD rapidly expands and ejects much of its hydrogen-rich matter. Once it regains thermal equilibrium, it contracts and becomes a luminous source of supersoft X-rays. This supersoft phase can last anywhere from days (for novae on massive WDs) to millions of years (for persistent supersoft sources). In this dissertation, we explore how the supersoft phase of accreting WDs proceeds and what observations of it reveal about the underlying WD. We present stellar models of persistent supersoft sources and novae. We then use these models to explain the isothermal nature of the ejecta as observed in the radio. We also use these models to test the efficacy of g-modes excited in the burning layer as an explanation of observed oscillations in the supersoft phase of novae. While we find excited modes, their periods are too short to account for the observed oscillations.

The advent of wide-field surveys has led to an exponential increase in the number of supernovae discovered each year. As the sample sizes of these objects have grown, we have discovered supernovae that show greater diversity within their spectroscopic classes than expected, as well as objects that do not fit into traditional classification schemes altogether. Studying these ``extreme" supernovae in greater detail is crucial to understanding the poorly-understood end stages of stars' lives; early-time observations within hours of the supernova explosion probe the progenitor star's stellar structure as well as its circumstellar environment, which can be used to test current theories of stellar evolution.

Here I present studies utilizing photometric and spectroscopic observations, primarily taken by Las Cumbres Observatory, of supernovae that occupy poorly-understood or unpopulated regions of parameter space. First, I examine an object with extreme ejecta velocities. Observations of this Type Ia supernova, SN 2019ein, within days of explosion show some of the fastest-moving ejecta of any supernova. The potential sources of this high-velocity ejecta in the context of the poorly-understood progenitor channels and explosion mechanisms of Type Ia supernovae are explored.

Next, I investigate the powering mechanisms and progenitor systems of supernovae with rapidly-evolving luminosities. High-cadence observations of objects within this region of supernova phase space reveal significant diversity in their circumstellar environments and powering mechanisms. In particular, I show evidence connecting luminous, rapidly-evolving unclassified transients with supernovae powered by interaction with circumstellar material. I also present the first sample study of a new class of supernovae, Type Icn supernovae, with rapidly-evolving light curves powered by interaction with circumstellar material that is both hydrogen- and helium-poor. Studying these unique and rare objects reveals that some are likely the explosions of stars less massive than expected from our current understanding of mass-loss in massive stars. Finally, I present evidence of diversity in the powering mechanisms and progenitors of the well-understood class of Type IIb supernovae. Each of these major findings challenges our understanding of supernova physics as well as theories of mass loss during the final stages of stellar evolution.

New wide-field and high-cadence transient surveys have enabled us to watch supernovae (SNe) from the moment of explosion. Paired with rapid and continuous monitoring facilities, these observations have revealed unprecedented features that bridge our understanding of their progenitor systems to explosion mechanisms. The Global Supernova Project (GSP) is a world-wide collaboration that uses Las Cumbres Observatory and various other ground- and space-based telescopes to study SNe from their discovery through all phases of follow-up. Here, I present three major advancements in core-collapse and superluminous SN observations from GSP and numerical modeling with MESA and STELLA. (1) SN 2018zd is a hydrogen-rich (Type II) SN discovered within a few hours of explosion. Its early ultraviolet color and narrow emission lines, along with light-curve morphology and nebular emission lines, are all consistent with an electron-capture SN explosion of a super-asymptotic giant branch star — the lowest-mass Type II SN. (2) Short-plateau SNe are luminous Type II core-collapse SNe with short light-curve plateaus, indicating significant pre-explosion mass loss resulting in partially stripped hydrogen-rich envelopes and early circumstellar material (CSM) interaction. Our light-curve model grid puts them in a rare transitional class from stripped red supergiants on the high-mass end of Type II SN progenitors. (3) Superlinear superluminous SNe (SLSNe) are Type II SLSNe with superlinear light curves powered by CSM interaction. Sample comparison and modeling suggest that the superlinear SLSNe are possibly compatible to thermonuclear SNe interacting with massive CSM which might be produced during common envelope evolution.

Many of the remaining uncertainties in stellar evolution can be addressed through immediate and long-term photometry and spectroscopy of supernovae. The early light curves of thermonuclear supernovae can contain information about the nature of the binary companion to the exploding white dwarf. Spectra of core-collapse supernovae can reveal material lost by massive stars in their final months to years. Thanks to a revolution in technology—robotic telescopes, high-speed internet, machine learning—we can now routinely discover supernovae within days of explosion and obtain well-sampled follow-up data for months and years. Here I present three major results from the Global Supernova Project at Las Cumbres Observatory that take advantage of these technological advances. (1) SN 2017cbv is a Type Ia supernova discovered within a day of explosion. Early photometry shows a bump in the U-band relative to previously observed Type Ia light curves, possibly indicating the presence of a nondegenerate binary companion. (2) SN 2016bkv is a low-luminosity Type IIP supernova also caught very young. Narrow emission lines in the earliest spectra indicate interaction between the ejecta and a dense shell of circumstellar material, previously observed only in the brightest Type IIP supernovae. (3) Type Ibn supernovae are a rare class that interact with hydrogen-free circumstellar material. An analysis of the largest-yet sample of this class has found that their light curves are much more homogeneous and faster-evolving than their hydrogen-rich counterparts, Type IIn supernovae, but that their maximum-light spectra are more diverse.