The COMET Experiment at J-PARC aims to search for the lepton flavour violating (LFV) process of muon-to-electron conversion in a muonic atom, µ − N → e − N, with a branching-ratio sensitivity of 6 × 10 −17 which is 4 orders of magnitude better than the present upper limit.Complemental searches for two kinds of muon LFV decay modes, µ + → e + γ and µ − N → e − N, are quite important in order to explore the parameter region predicted by most well-motivated theoretical models beyond the Standard Model such as SUSY-GUT, seesaw, little-Higgs, extradimension, etc. MEG collaboration reported the latest result on µ + → e + γ search at PSI in Switzerland, and made a strong limit on new physics models, i.e. filling in the missing peace of muon LFV, µ − N → e − N, will certainly play a very important role. The need for such an unprecedented sensitivity places several stringent requirements on both the muon beam and the detector system. Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).