The standard model of particle physics is an extremely successful theory of fundamental interactions, but it has many known limitations. It is therefore widely believed to be an effective field theory that describes interactions near the TeV scale. A plethora of strategies exist to extend the standard model, many of which contain predictions of new particles or dynamics that could manifest in proton-proton collisions at the Large Hadron Collider (LHC). As of now, none have been observed, and much of the available phase space for natural solutions to outstanding problems is excluded. If new physics exists, it is therefore either heavy (i.e. slightly above the reach of current searches) or hidden (i.e. currently indistinguishable from standard model backgrounds). We summarize the existing searches and discuss future directions at the LHC.