The molecular self-diffusion coefficients in nematic and smectic-A thermotropic liquid crystals are measured using stimulated-echo-type 2H and 1H pulse-field-gradient spin-echo nuclear magnetic resonance (PGSE NMR) combined with multiple-pulse dipolar decoupling and slice selection. The temperature dependence of the principal components of the diffusion tensor in the nematic phase follows a simple Arrhenius relationship except in the region of nematic-isotropic phase transition where it reflects, merely, the decrease of the molecular orientational order. The average of the principal diffusion coefficients in the isotropic-nematic phase transition region is close to the diffusion coefficient in the isotropic phase. At the nematic-smectic-A phase transition the diffusion coefficients change continuously. The results in nematic phase are best described in terms of the affine transformation model for diffusion in nematics formed by hard ellipsoids. In the smectic-A phase the data are interpreted using a modified model for diffusion in presence of a periodic potential along the director.