Muonium (Mu), may be regarded as a radioactive hydrogen atom with a positive muon as its nucleus, and is formed in a range of media which are irradiated with positive muons. This exotic atom can be considered as a second radioisotope of hydrogen, along with tritium. Addition of this light atom (with a mass 1/9th that of a normal hydrogen, protium, atom) to unsaturated organic molecules forms free radicals, in which the muon serves as a radioactive and magnetic probe of their kinetic and structural properties. Suitable examples are chosen to illustrate the very large functionality of organic radicals which have been measured using muons and various methods of μSR, where μ stands for muon, S for spin and R may refer to rotation, resonance or relaxation. The principal techniques illustrated are transverse-field muon spin rotation (TF-μSR), avoided level crossing muon spin resonance (ALC-μSR) and longitudinal-field muon spin relaxation (LF-μSRx). Structural studies of radicals, the determination of mechanisms for radical formation, the measurement of radical stabilisation energies, the determination of the kinetics of reactions of free muonium atoms and of free radicals have all been accomplished using TF-μSR methods. It is further shown that TF-μSR is most useful in measuring radical reaction rates in non-aqueous media, to provide information of relevance to cell membrane damage and repair. Muonium may further be used as a mechanistic probe since it determines a true pattern of H-atom reactivity in molecules, against which results from similar radiolysed materials may be compared. [In many solid materials that are exposed to ionising radiation, apparent H-atom adduct radicals are detected but which originate from charge-neutralisation of positive holes (radical cations) and ejected electrons, without free H-atoms being formed. DNA is the superlative example of this. Free H-atoms normally feature in the province of radiolysed aqueous media]. The applications of ALC-μSR and LF-μSRx in studying the reorientation of reactive radicals on reactive surfaces forms the substantive proportion of the review: considered specifically are radicals sorbed in zeolites, in clays and in porous silica, in porous carbons and on ice-surfaces, in connection with their role as intermediates in catalytic systems, particularly hydrocarbon cracking and oxidation processes, and in atmospheric aerosol chemistry. The formation of muonium and other muon species in cation-exchanged zeolite-X samples are also considered, according to the evidence of longitudinal field repolarisation measurements. Finally, mention is given of the use of μSR techniques for studying radicals in the gas-phase.