We explain the origin of the puzzling high frequency peaks in the variability power spectra of accreting neutron stars and black holes as a non-linear 1:2 or 1:3 resonance between orbital and radial epicyclic motion. These resonances are present because the gravitational field deviates strongly from a Newtonian 1/r potential. Our theory agrees with the recently reported observations of two QPOs, at 300 Hz and 450 Hz, in the black hole candidate GRO J 1655-40.

PACS numbers: 97.60.Lf, 97.10.Gz, 97.10.Sj, 97.60.Jd