Sum-over-states density functional perturbation theory (SOS-DFPT) has been used to calculate C-13, N-15, and O-17 NMR chemical shifts of 20 molecules, for which accurate experimental gas-phase values are available. Compared to Hartree-Fock (HF), SOS-DFPT leads to improved chemical shift values and approaches the degree of accuracy obtained with second order Moller-Plesset perturbation theory (MP2). This is particularly true in the case of N-15 chemical shifts where SOS-DFPT performs even better than MP2. Additional improvements of SOS-DFPT chemical shifts can be obtained by empirically correcting diamagnetic and paramagnetic contributions to compensate for deficiencies which are typical of DFT.