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The quest for $μ\to e γ$: present and future

The quest for $μ\to e γ$ is one of the most important endeavors to search for New Physics beyond the Standard Model. In this talk I will review the current status of the experimental searches by the MEG Collaboration at PSI. I will also present a study of the experimental limiting factors that will define the ultimate performances, and hence the sensitivity, in the search for $μ\to e γ$ with continuous muon beams of extremely high rate (one or even two orders of magnitude larger than the present beams), whose construction is under consideration for the next decade.