A formulation for the calculation of nuclear magnetic resonance (NMR) shielding tensors, based on density functional theory (DFT), is presented. Scalar-relativistic and spin-orbit coupling effects are taken into account, and a Fermi-contact term is included in the NMR shielding tensor expression. Gauge-including atomic orbitals(GIAO) and a frozen-core approximation are used. This formulation has been implemented, and H-1 and C-13 NMR shifts of hydrogen and methyl halides have been calculated and show good agreement with experiment. C-13. NMR shifts of 5d transition metal carbonyls have been calculated and show improved agreement with experiment over previous scalar-relativistic calculations. For the metal carbonyls it is shown explicitly that the combination of spin-orbit coupling and magnetic field mixes spin triplet states:into the ground state, inducing a spin density that then interacts with the nuclei of the metal carbonyl via the Fermi-contact term. Results indicate that the; Fermi-contact contribution to the C-13 NMR of the metal carbonyl ions increases with increasing oxidation state of the ion. It is reasoned that as the oxidation state increases,:rr back bonding decreases and sigma bonding increases, within the metal-carbon bond, thus facilitating a greater transfer of spin density from the metal to the carbon nucleus, and thus increasing the Fermi-contact contribution to the NMR shielding of the carbon.