The upcoming LHC runs are an unparalleled opportunity for particle physics. The discovery of the Higgs boson almost 50 years after its theoretical prediction has finally established the mechanism of electroweak symmetry breaking. It has revealed the final missing puzzle piece of the Standard Model and established quantum field theory as the mathematical description of fundamental interactions. Nevertheless, much about the Standard Model remains mysterious. While it describes experiments with uncanny precision, it cannot serve as a complete description. It fails to describe neutrino masses, dark matter, the matter-anti-matter asymmetry of the Universe, the approximate gauge coupling unification, and inflation. In addition, many of its structural features such as the different gauge symmetries, the matter content, its patterns of masses and mixings, and the absence of strong CP-violation hint at underlying structures at higher energies. There are many compelling visions for how these mysteries might be linked to new structure at the TeV scale, within reach of the upgraded LHC.