When polarized positive muons stop in a solid, they may be depolarized prior to their decay. The presence of a magnetic field along the direction of initial polarization will inhibit, or quench, some of this depolarization. We have studied depolarization and quenching in a variety of solids, including semiconductors, at temperatures as low as 4.2\ifmmode^\circ\else\textdegree\fi{}K. No evidence has been found that any muon depolarization is associated with the formation of a muonium atom which is bound for >${10}^{\ensuremath{-}10}$ sec, even in samples at 4.2\ifmmode^\circ\else\textdegree\fi{}K. For many samples and temperatures, the quenching of the observed depolarization is consistent with a model in which the muon captures and loses electrons repeatedly, forming a succession of briefly bound muonium atoms. In some of the semiconductors at low temperatures, as well as in other samples, a large fraction of the depolarization is quenched by a field of 100 G. This is seen as evidence for another depolarization mechanism. We also find that the time during which depolarization occurs in boron carbide decreases with decreasing temperature.