Nuclear magnetic resonance (NMR) measurements on alkali fullerides AxC60 (A = K, Rb, Cs; x = 0, 1, 3, 4, 6) are reviewed, with an emphasis on the interpretation of data on the superconducting A3C60 compounds. It is argued that C-13 NMR spectra and spin-lattice relaxation rates in the A3C60 compounds are most strongly affected by electron-nucleus hyperfine couplings of the dipolar, rather than Fermi contact, type and that antiferromagnetic electron spin fluctuations, structural disorder, or a combination of these two factors may influence the measurements substantially. Interpretation of the NMR data in terms of electronic densities of states is therefore not straightforward. C-13, Rb-87 and Cs-133 NMR measurements on A1C60 compounds provide information about the phase transitions and electronic properties of these materials. All A1C60 compounds have structural phase transitions in the 270-420 K range. The high temperature phases are paramagnetic insulators. Below their respective phase transitions, Rb1C60 and Cs1C60 are single-phase compounds with greatly reduced spin susceptibilities, while K1C60 is a phase-separated mixture of K3C60 and KdeltaC60 with delta much less than 1. C-13 spin-lattice relaxation measurements on Rb4C60 show non-metallic behavior, but the relaxation rates are anomalously large. Relaxation measurements on A6C60 compounds are consistent with electronic band structure calculations, which show them to be diamagnetic insulators.