The nuclear quadrupole coupling constants (NQCCs) for the nitrogen and oxygen nuclei in N2O have been determined using a variety of computational methods (MP2, QCISD, DFT with B3LYP, PBE0, and B3PW91 functionals, CCSD, CCSD(T), CASSCF, and MRCI) combined with correlation-consistent basis sets. When compared to the available experimental determinations, the results demonstrate that only CCSD(T) and MRCI methods are capable of accurately predicting the NQCCs of the central and terminal nitrogen atoms. The spin-rotation and magnetic shielding tensors have also been determined and compared to experimental measurements where available. N-14 and O-17 NMR relaxation data for N2O in the gas phase and a variety of solvents is reported. The increase in the ratio of N-14 spin-lattice relaxation times in solvent for the central and terminal nitrogens supports previous reports of the modification of the electric field gradients at these nuclei in van der Waals complexes. Ab initio computations for the linear FH center dot center dot center dot N2O complex confirm the large change in EFGs imposed by a single perturber.