The Large Hadron Collider has thrust humanity into the TeV-scale era of particle physics, and with it we have discovered a new Higgs-like particle and set stronger limits on possible theories of supersymmetry and beyond-the-Standard-Model physics. But while the experimental community continues to improve their searches for new physics, the lack of clear beyond-the-Standard-Model discoveries so far has made it apparent that if we are to discover new physics there, the theoretical community must continually develop new methods as well. New tools must be forged and applied to the problems we face at the Large Hadron Collider and possible future particle colliders. This dissertation expresses one contribution, out of many, in this regard. We begin by using boosted top tagging to look for signals of R-parity conserving supersymmetry. Then we use other jet substructure techniques to look for signals of Rparity violating supersymmetry. Lastly, we use the SUSY-Yukawa sum rule to look beyond the phenomenology and probe the crucial cancellation of divergences at the heart of supersymmetric theories.