We have determined the principal elements of the chemical shift tensors for a series of metal-olefin complexes: [Ag(cod)(2)]BF4(cod = cis,cis-cycloocta-1,5-diene), [CuCl(cod)](2), PtCl2(cod), [RhCl(cod)](2), and K[PtCl3(C2H4)] using magic-angle sample spinning and a Bayesian probability method to deduce mu, rho in the Herzfeld-Berger equations. These principal elements have also been computed by using density functional methods with two different types of functionals and partial geometry optimization. The overall slope and R-2 values between the theoretical and experimental tensor elements are good, ranging from 1.06 to 1.16 for the slope (versus the ideal value of 1) and 0.98-0.99 for the goodness of fit parameter R-2 (versus the ideal value of 1). The use of a hybrid functional results in a slightly worse slope, an effect which is largest for the compounds with the largest paramagnetic shifts. There are no particularly good correlations between C-C bond lengths, isotropic/anisotropic shift tensor elements or computed bond orders; however, the correlation between shielding and (Mulliken) charge of similar to-120 ppm/electron is consistent with previous experimental estimates on olefins and aromatic compounds. The orientations of the shielding tensor elements in the cod complexes change in a relatively continuous manner with increases in shielding (from d(10) to d(8) metals), with delta(33) becoming rotated (37.5 degrees) from the normal to the C=C bond axis in [RhCl(cod)](2). Overall, these results indicate that density functional methods permit the relatively accurate reproduction of metal-ligand shielding patterns in systems whose structures are known, which should facilitate their use in probing metal-ligand geometries in systems whose structures are less certain, such as in metalloproteins.