Classically periodic molecular vibration (such as a totally symmetric stretch) can be unstable against the addition of small components of other modes, depending on anharmonic coupling strengths, near resonance of fundamental frequencies, and the total energy. We report here on some very strong correspondances between classical stability of the motion and quantum spectral features, wave functions, and energy transfer. The usual concept of a vibrational Fermi resonance turns out to apply best to the case where the transition to classical instability occurs at an energy below the first resonant quantum levels (this is the case for the famous Fermi resonance in CO2). In the (probably more common) event that resonant classical instability should set in above several quanta of energy in the mode of interest, the quantum spectrum shows tell‐tale pre‐ and post‐resonant signatures which include a t t r a c t i o n of quantum levels (rather than the usual Fermi repulsion) and other features not normally associated with Fermi resonances. Evidence is presented which suggests that certain types of periodic motion in anharmonic molecules act as ’’traps’’, and are resistant to energy exchange with other types of motion. Numerical evidence linking the classical and quantum behavior, together with a new semiclassical theory presented here provides a very explicit connection between quantum and classical anharmonic motion.