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The search for charged lepton flavour violation (CLFV) has enormous discoverypotential in probing new physics Beyond the Standard Model (BSM). Among themuonic CLFV processes, $\mu \to e$ conversion is one of the most importantprocesses, having several advantages compared to other such processes. Wedescribe the COMET experiment, which is searching for $\mu \to e$ conversion ina muonic atom at the J-PARC proton accelerator laboratory in Japan. The COMETexperiment has taken a staged approach; the first stage, COMET Phase-I, iscurrently under construction at J-PARC, and is aiming at a factor 100improvement over the current limit. The second stage, COMET Phase-II is seekinganother 100 improvement (a total of 10,000), allowing a single eventsensitivity (SES) of $2.6 \times 10^{-17}$ with $2\times 10^{7}$ seconds ofdata-taking. Further improvements by one order of magnitude, which arise fromrefinements to the experimental design and operation, are being consideredwhilst staying within the originally-assumed beam power and beam time. Such asensitivity could be translated into probing many new physics constructions upto $\mathcal{O}(10^{4})$ TeV energy scales, which would go far beyond the levelthat 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.

COMET - A submission to the 2020 update of the European Strategy for Particle Physics on behalf of the COMET collaboration