I review the motivation for and status of searches for violation of muon and electron number conservation, concentrating on muon initiated processes. I then discuss the progress expected in the next few years, concentrating on a new experiment, E940 at BNL (the MECO experiment). It will improve the experimental sensitivity for the process μ^-Ν —> e^-N to below 10^-16 , roughly 4 orders of magnitude better than the current limit.