The SINDRUM II experiment at PSI searches for the coherent conversion of a muon into an electron in titanium. The theoretical motivation, the SINDRUM II detector and the results of the first data‐taking period in 89 are presented. In total 3.2 million events have been analyzed and no candidate for the process μ−Ti→e−Ti has been found. An upper limit of Bμe<4.4⋅10−12 (90% C.L.) is obtained for the branching ratio. The plans to lower the sensitivity by two orders of magnitude are discussed. A muonium‐antimuonium conversion experiment is being set up at PSI using the refurbished SINDRUM I detector. The goal of the experiment is to improve the sensitivity for the effective coupling constant GMM by a factor of ∼100.