  We begin this thesis with an extensive pedagogical introduction aimed at
clarifying the foundations of the hierarchy problem. After introducing
effective field theory, we discuss renormalization at length from a variety of
perspectives. We focus on conceptual understanding and connections between
approaches, while providing a plethora of examples for clarity. With that
background we can then clearly understand the hierarchy problem, which is
reviewed primarily by introducing and refuting common misconceptions thereof.
We next discuss some of the beautiful classic frameworks to approach the issue.
However, we argue that the LHC data have qualitatively modified the issue into
`The Loerarchy Problem'---how to generate an IR scale without accompanying
visible structure---and we discuss recent work on this approach. In the second
half, we present some of our own work in these directions, beginning with
explorations of how the Neutral Naturalness approach motivates novel signatures
of electroweak naturalness at a variety of physics frontiers. Finally, we
propose a New Trail for Naturalness and suggest that the physical breakdown of
EFT, which gravity demands, may be responsible for the violation of our EFT
expectations at the LHC.
