Pulsed NMR measurements were performed at 10 and 20 MHz on thin 3He films on graphitic surfaces in the temperature range between 0. 35 and 4.2 K. Most of the measurements reported here have been obtained with basal-planeoriented graphite (Grafoil) outgassed izt 130~ but in some experiments graphitized carbon black powder (Sterling b-T), vacuum-baked at 1100~ was used for comparison. The aHe coverages examined range from O. 1 to 80 monolayers on Grafoil and from 0.3 to 1.0 monolayers on Sterling FT. Measurements have also been made on 3He-4He films on Grafoil with one-monolayer quantity of 3He mixed with various amounts of 4He. The measured free-induction decay time T~, the spin-echo decay time T+2, and the spin-lattice relaxation time T1 suggest that the observed relaxation phenomena in 3He are largely governed by the 3He-substrate interaction. In 3 "~ r the case of He on Grafoil, T2 is much longer than T2, evidently as a result of large local magnetic field gradients and significant lateral mobility of 3He on this substrate. These results, in con]unction with a simple model of spin diffusion in restricted ~eometries, lead to values of the diffusion coefficient Ds of about 2.1 xlO-Scm /sec at 4.2K and 7.0xlO-6cm2/sec at 1.2K for one-monolayer coverages; these values of Ds are indicative of a nonlocalized 3He system. To verify the model used, the restricted diffusion analysis was applied to room-temperature measurements on C6F6 in Grafoil ; the values of the diffusion coefficient obtained in this way are in good agreement with diffusion rates measured directly in bulk liquid C6F6 at room temperature. In 3 t the case of He on Sterling FT, the restricted diffusion analysis of the T2 data gives Ds-~2 • lO-S cmZ/sec for 0.6 monolayer of 3He; this value of Ds suggests a relatively localized system. The measured T'2 is found to decrease monotonically with coverage and there is no evidence of any phase transition in 3He films for coverages up to one monolayer.