Sum-over-states density functional perturbation theory (SOS-DFPT) and Hartree-Fock (HF) theory based on the "individual gauge for localized orbitals" (IGLO) scheme were used to calculate C-13,N-14/N-15, and O-17 NMR chemical shifts of 23 molecules. Employing the (11s7p2d/6s2p)[7s6p2d/4s2p] basis set, SOS-DFPT chemical shifts were obtained that are clearly better than HF- or UDFT-based chemical shifts, as is reflected by mean absolute deviations of 2.8 (SOS-DFPT), 7.6 (UDFT), and 5.6 ppm (HF) for C-13 chemical shifts, 11.8, 22.1, and 100.5 ppm for N-14/N-15 chemical shifts, and 36.4, 57.2, and 45.7 ppm for O-17 chemical shifts with regard to experimental values. By estimating appropriate gas phase values for N-14/N-15 chemical shifts from measured solvent effects, average errors of SOS-DFPT chemical shifts are reduced to 3 ppm, which is close to uncertainties in experimental values. SOS-DFPT is the method of choice for predicting reasonably accurate NMR chemical shifts at relatively low computational cost even in cases where correlation corrections are large. However, further improvements of SOS-DFPT are necessary to calculate more reliable O-17 shift values of conjugated pi-systems.