  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 $2\times 10^{7}$ 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.
