The past ~ 50 years have seen a remarkable success of particle physics. In the 1970s, the Standard Model was formulated and in 2012 its final ingredient, the Higgs boson, was discovered at the Large Hadron Collider (LHC). The Standard Model describes virtually all particle physics observable in the laboratory. However, despite this success, the Standard Model has a number of shortcomings. Some problems stem from its mathematical structure, most famously the hierarchy problem. Further, the Standard Model fails to describe the composition of our Universe, for example, it cannot explain the observed Dark Matter. Thus, the need for physics beyond the Standard Model is clear. A long series of experiments has been conducted to search for this new physics. Alas, these experiments came up empty handed.This thesis discusses two lines of work: 1) Arguably, the Higgs sector of the Standard Model is its least constrained part and simultaneously intimately related to many of the Standard Model's shortcomings. We discuss models extending the Higgs sector, both in a general and in a supersymmetric setting, and how they can be probed at the LHC. 2) A century after the first evidence for Dark Matter emerged, we still don't know what it is made up of. We discuss some models for Dark Matter, including axions and a particular model for Weakly Interacting Massive Particle (WIMP) Dark Matter. Then, we present some methods to search for WIMP Dark Matter, focusing on paleo-detectors, a proposed method where one would search for the traces of WIMP-nucleus interactions left in ancient minerals.