Several aspects of muon trapping in metals have been studied during the last two years, but the situation is still far from clear. The precise nature of the traps as well as the mechanisms leading to trapping seem to require more detailed investigations than those carried out so far. The present review contains therefore a certain number of ideas which should be regarded as working hypotheses rather than established facts or descriptions of positive muon behaviour. The muon depolarization at low temperatures (between 0.03 and 2 K) has been reviewed by Hartmann [i]. These new experiments show, at least for muons in aluminium, that the thermal excitation of the metal latti~has a strong influence on the muon mobility already at temperatures less than 0.i K. The first diffusion steps may be fast enough that we observe practically all muons in some kind of impurity trapped state, without knowing their immediate pre-history. Low-T experiments may provide information on the starting conditions for diffusion. These are, of course, a necessity for the development of any model for "diffusion toward impurities" or "diffusion limited by impurities". We will first consider muons in f.c.c, metals. The situation in the last stages of the slowingdown process in metals has recently been considered by Browne and Stoneham [2], introducing the possibility of stopping (i.e. reaching a self-trapped state) preferentially in the strain field of impurities, as contrasted to the idea of random initial stopping in the lattice. The recent low-temperature experiments in doped A1 indicate, however, that the initial self-trapping is random and that regions close:to impurities are reached only by diffusion. This happens both below 0.i K and around I0 K when the impurity concentration is large enough (around i00 ppm). In an intermediate temperature range (around 1 K) the muons are mobile, but diffusing only over a limited region of space, out of range for the traps. The general conditions for self-trapping have also been reviewed by Stoneham and collaborators [2,3]. Self-trapping at low temperatures should be determined by a band-width (equivalent to a tunneling matrix element J) in the non-trapped state and the energies - thermal and/or elastic - needed to overcome a certain selftrapping barrier. Considering the relevant magnitudes (including lattice zeropoint energy) it is most likely that self-trapping indeed occurs in a strainfree lattice for the f.c.c, metals (J ~ i0 - i00 ~eV), but perhaps not in the b.c.c, metals (J ~ 1 meV). The initial situation before diffusion can therefore tentatively be described as follows: the muons come to rest and get self-trapped at random positions in the lattice. The possibility for a coherent tunneling or "band motion" in the selftrapped state, that is of the muon plus the surrounding deformation, exists but is strongly reduced and is described by the effective tunneling matrix element Jeff = J exp[S(T)], With Jeff's of the order of ~eV this latter process is extremely sensitive to the distribution of strains and is also limited to a lowtemperature region, as visualized in the strain-temperature (or static-dynamic disorder) "phase diagram" of Fig. i.