Muon decay in the Standard Model requires the emission of two neutrinos thanks to the conservation of Lepton Flavour. Given neutrino oscillations, however, this conservation is known to be violated in the Standard Model. The COMET experiment is one of a handful of projects hoping to demonstrate Charged Lepton Flavour Violation, searching for Coherent Muon to Electron Transitions, where a muon converts to an electron in the presence of an atomic nucleus, without neutrino emission. I present here an overview of muon-to-electron conversion and the COMET experiment itself, with which the UK has been involved since its inception. Currently under construction to begin Phase-I data taking in JFY 2018, the first stage will see a factor 100 improvement on the current limit to around 3x10^-15. Phase-II will then go a further two orders of magnitude running early next decade. Setting such stringent limits comes with significant design challenges. These, their solutions, and recent developments in facility, beam-line and detector construction will also be presented.