This Ph.D. thesis presents the results of magnetooptical spectroscopic investigations of free radicals isolated in Ar matrices at cryogenic temperatures (≤20 K). The experiments required development of a high-resolution simultaneous magnetic circular dichroism (MCD) and absorption spectrometer, MOD4. Design, construction and commission of MOD4 comprised a significant portion of this project, and its description constitutes a substantial part of the 'Experimental' chapter of the thesis. The results of matrix-isolation experiments on the first-row radical monohydrides, CH, NH (and ND) and OH (and OD), and the theoretical analyses of the data comprise the main body of this thesis, and are presented in three chapters. Temperature and magnetic-field dependencies of the MCD and absorption over the ranges T≈1.4 –17 K and B = 0 to ∼ 4.5 T, are interpreted in terms of a spin-orbit (SO) –crystal-field (CF) model. The data are analysed by the application of quantum-mechanical and group-theoretical techniques, utilising moment analysis and (in the case of NH) spectral simulation. These allow SO and CF parameters to be extracted. Assignments of structure observed in the spectra have also been attempted. Radical monoanions and monocations of buckminsterfullerene (C₆₀) have been investigated in Ar matrices using MCD and absorption spectroscopy, over the ranges T≈1.6 –30 K and B = 0 –4 T. Preliminary conclusions have been drawn from these data, concerning the symmetry of the molecules, but more experimental and theoretical work needs to be done. Ferricenium, the radical cation of ferrocene, has been isolated in Ar (with a SF6 counter-ion) for the first time. Preliminary MCD and absorption data are presented, and show well-resolved vibronic structure; assignments to totally symmetric and Jahn-Teller-active vibrational overtones are made.