For the first time, an NMR spectrometer equipped with a high pressure probe has been interfaced with a Vapor-Liquid-Equilibrium apparatus to study solvent-solute interactions in supercritical fluids. The following three paragraphs detail the sequential progress of the work where we focused on the pure solvent, the pure solvent near its critical point and then the mixtures. Spin-lattice relaxation time T\sb1 and self-diffusion coefficient D in \sp13CO\sb2 have been measured from 0 to 75\sp∘C at pressures up to 500 bar. The governing relaxation mechanism in this range is shown to be spin-rotation relaxation. For both T\sb1 and D data, a kinetic theory based model describes well the low density values, whereas a hydrodynamics based model works adequately at high densities. Using recent molecular dynamics calculations, we found that the smooth hard-sphere theory predicts surprisingly well the self-diffusion of CO\sb2 at densities above critical. T\sb1 and D have then been measured near the critical point of CO\sb2. T\sb1 values are unprecedented and this is the third D determination in the critical region. The two previous D determinations are in serious disagreement. One reported the presence of a strong critical anomaly whereas the other observed that D behaves normally in the critical region. No critical anomaly was found for either T\sb1 or D. Finally, T\sb1 and D of \sp13CO\sb2 and C\sb16\sp1H\sb34 have been measured on relevant isotherms from atmospheric pressure up to the critical point of the mixture in the coexisting vapor and liquid phases. The data have been correlated with phase compositions determined earlier in this laboratory and with calculated viscosities. \sp13CT\sb1 data indicate that the reorientational correlation time of the CO\sb2 molecule is constant in the liquid phase up to the critical region. \sp1HT\sb1 data are in qualitative agreement with theory. The ratios of the self-diffusion coefficients in each phase is closely related to the ratios of the partial molar volumes, in agreement with arguments from statistical mechanics and irreversible thermodynamics. Calculated mutual diffusivities are found to be in excellent agreement with available experimental diffusivities.