mu SR has been used to study a variety of polymers with very different electronic properties. In conducting polymers, the muon-generated radical states take the form of highly mobile polarons. Muon spin relaxation has been used to study the mobility of these polarons and to measure the temperature dependence of their intra-chain and inter-chain diffusion rates. it is found that the transport properties are strongly influenced by the librational ring modes of the phenylene rings in these polymers. In contrast, the muon-generated radical states in non-conducting polymers such as polybutadiene remain localised near the site of the muon. High field muon spin rotation, avoided level crossing resonance and longitudinal relaxation studies have been made, using the muon radical state as a probe of the dynamical properties of the polymer. Dramatic changes in the mu SR signals are seen on going through the glass-rubber transition, as various dynamical degrees of freedom become frozen out. Additional information about the stability of the muon radical states on the microsecond timescale has also been obtained using RF muon spin rotation techniques. Using time-delayed RF resonance of the diamagnetic state at the RIKEN-RAL muon facility, the transition rate between paramagnetic and diamagnetic states could be studied as a function of temperature. (C) 2000 Elsevier Science B.V. All rights reserved.