Calculations of 77Se nuclear magnetic resonance (NMR) chemical shifts and shieldings are reported for a number of selenium containing compounds. The calculated shifts span a range of about 2,800 ppm, and cover therefore almost the complete range of known 77Se chemical shifts. The calculations are based on density functional theory (DFT); "gauge including atomic orbitals" (GIAO's) are used. The DFT-GIAO method is tested against other theoretical approaches. It is able to predict chemical shifts as well as individual tensor components of the shielding tensor with about the same quality as sophisticated ab initio methods. DFT is even capable of handling highly correlated systems like Se42+. Other methods fail in this particular case.