The search for charged lepton flavour violation (CLFV) has enormous discovery potential in probing new physics Beyond the Standard Model (BSM). Among the muonic CLFV processes, $mu to e$ conversion is one of the most important processes, having several advantages compared to other such processes. We describe the COMET experiment, which is searching for $mu to e$ conversion in a muonic atom at the J-PARC proton accelerator laboratory in Japan. The COMET experiment has taken a staged approach; the first stage, COMET Phase-I, is currently under construction at J-PARC, and is aiming at a factor 100 improvement over the current limit. The second stage, COMET Phase-II is seeking another 100 improvement (a total of 10,000), allowing a single event sensitivity (SES) of $2.6 times 10^{-17}$ with $2times 10^{7}$ seconds of data-taking. Further improvements by one order of magnitude, which arise from refinements to the experimental design and operation, are being considered whilst staying within the originally-assumed beam power and beam time. Such a sensitivity could be translated into probing many new physics constructions up to $mathcal{O}(10^{4})$ TeV energy scales, which would go far beyond the level that can be reached directly by collider experiments. The search for CLFV $mu to e$ conversion is thus highly complementary to BSM searches at the LHC.
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