Recent progress is surveyed in regard to the importance of molecular species containing unpaired electrons in catalytic systems, as revealed using ESR spectroscopy. The review begins with studies of enzymes and their role directly in biological systems, and then discusses investigations of various artificially created catalysts with potential human and environmental significance, including zeolites. Among the specific types of catalytic media considered are those for photocatalysis, water splitting, the degradation of environmental pollutants, hydrocarbon conversions, fuel cells, ionic liquids and sensor devices employing graphene. Studies of muonium-labelled radicals in zeolites are also reviewed, as a means for determining the dynamics of transient radicals in these nanoporous materials.