Relativistic and substituent effects on C-13 NMR chemical shifts in mercurimethanes and on H-1 shifts in organomercury hydrides have been studied by density functional calculations, comparing quasirelativistic and nonrelativistic effective-core potentials for mercury. The positive shift increments in the C-13 shifts as a function of HgCl or HgCN substituents in the mercurimethanes CHn(HgX)(4-n)(X = Cl, CN; n = 0-4) are due to scalar relativistic effects. The relativistic effects for a given structure and the influence of the relativistic Hg-C bond contraction partly oppose each other, in contrast to results obtained recently for O-17 shifts in oxo complexes. These differences are due to different types of metal orbitals involved in bonding, mainly of 6s-character for the mercury compounds but predominantly of 5d-character for the oxo complexes. Remaining discrepancies between computed and experimental C-13 shifts of CH3HgX for more electropositive substituents X = CH3, SiH3 and particularly between computed and experimental H-1 shifts in organomercury hydrides RHgH (R = CH3, C2H5, C2H3, C6H5, C6F5), appear to be largely due to spin-orbit coupling, as indicated by preliminary calculations of spin-orbit corrections to the chemical shifts. The spin-orbit contributions are almost entirely due to a rho(u)(X-Hg-Y) --> pi*(Hg 6(x,y))-type coupling and affect exclusively the shift tensor components perpendicular to the X-Hg-Y axis. The magnitude of the spin-orbit corrections correlates well with the inverse of the energy differences between the corresponding Kohn-Sham MOs. Thus spin-orbit coupling probably accounts in part for the increase of the C-13 shifts in CH3HgX with decreasing electronegativity of X, and for similar trends of the H-1 shifts in organomercury hydrides. In addition to the chemical shift results, analyses of the molecular and electronic structures of the mercurimethanes reveal interesting counterexamples to Bent's rule.