The unusual methods of preparation and analysis of spin polarization in mSR spectroscopy, which exploit the unique properties of the positive muon, are introduced in this article. Following a summary overview of applications, particular attention is paid to the problem of spin-lattice relaxation for a muon experiencing a hyperfine interaction with a single unpaired electron. The specific cases considered are the interstitial diffusion of muonium-the 1-electron atom which may be considered as a light isotope of hydrogen-and the molecular dynamics of organic radicals labelled by muonium. Rate equations for the evolution of population in the hyperfine-coupled spin states are solved numerically for various relaxation mechanisms. The formalism is equally valid for conventional ESR studies of paramagnetic states but is pursued specifically to simulate T -relaxation in mSR. The simulations are compared with literature data. Also treated is the case of intermittent 1 hyperfine coupling, appropriate to electron capture and loss in semiconductors or soliton motion in polymers; for this, a ลฝ Monte Carlo approach is used to simulate the muon response. For low-dimensional motion, the relaxation function is not . exponential, so that a unique value of T cannot be defined. Finally, a proposal is made to implement muon-T 1 1 r measurements in the rotating frame; this is designed for the selective study of electronically diamagnetic muonium states ลฝ . i.e., those without hyperfine coupling in the presence of a paramagnetic muonium or radical fraction. q 1998 Elsevier Science B.V.