Most searches for new physics at the Large Hadron Collider assume that a new particle produced in pp-collisions decays almost immediately, or is non-interacting and escapes the detector. However, a variety of new physics models predict particles which decay inside the detector at a discernible distance from the interaction point. Such long-lived particles would create spectacular signatures and evade many prompt searches. In this talk I will focus on a search for long-lived particles in events with a displaced vertex and a muon. I will also discuss challenges for the Muon Spectrometer in the face of increasing LHC luminosity.