Particle Physics in the LHC Era

During the past 100 years experimental particle physicists have collected an impressive amount of data. Theorists have also come to understand this data extremely well. It was in the first half of the 20th century the efforts of the early pioneers of quantum mechanics laid the ground work for this understanding: quantum field theory. Through the tireless efforts of researchers during the later half of the 20th century many ideas came together to form what we now call the Standard Model (SM) of particle physics. Finally, it was through the ideas of the renormalization group and effective field theory that the understanding of how the SM fits into a larger framework of particle physics was crystallized. In the past four years the Large Hadron Collider (LHC) has made more precise measurements than ever before. Currently the SM of particle physics is known to have excellent agreement with these measurements. As a result of this agreement with data, the SM continues to play such a central role in modern particle physics that many other theories are simply known as `Beyond the Standard Model' (BSM) as we know any new models will simply be an extension of the SM. Despite agreement with experiment, the SM does suffer from several shortcomings that raise deeper questions. In this dissertation we study models that address the two of the outstanding theoretical problems of the SM - the Strong CP Problem and the fine tuning of the Higgs mass. We study models that solve or ameliorate these problems, and their implications for collider physics and astrophysics.