Experiments with muons are a unique window to investigate the Standard Model of particle physics and to look for New Physics effects with high sensitivity. The search for the lepton flavor violating $\mu ^+\rightarrow e ^+\gamma$ decay will reach a sensitivity of few 10$^{-14}$ within the next three years thanks to the MEG II experiment, presently taking data at the Paul Scherrer Institut, with a beam intensity of 4 $\times$ 10$^{7}$ muons/s. On the other hand, there is a world-wide effort to increase the muon beams intensity at present facilities by a factor 100. In order to exploit such intense muon beams a new generation of experiments has to be
designed, since the present experimental concept is not adequate to deal with the expected increase
in rate, radiation and background. An international study group has been recently set, the Study Group for Future $\mu \rightarrow e\gamma$ search experiments, to organize this effort. This paper reviews the present main experimental directions identified, the results of simulations and R&D studies made so far.