E.S.R. spectra of phosphorescent aromatic molecules in frozen solution present, forg≲ 2, a line arising from absorption of two microwave quanta. We show how the resonance field may be computed for any molecular orientation. We find the value of the stationary resonance field and we demonstrate that the transition probability goes through a maximum for the same field value. We give a perturbation treatment followed by an exact derivation of this property. When the electron spin couples strongly enough to the nuclei, we record hyperfine structure in the two-quantum peak. We also analyse this phenomenon theoretically. For an axial hyperfine tensor we are able to account for the observed lineshape, thereby confirming our treatment of the stationary resonance.