We consider phenomena which are frequently designated jointly as the Jahn--Teller effect and result from dynamic electron-vibrational (vibronic) coupling in polyatomic systems with degenerate and pseudodegenerate terms. Principal attention is paid to the case of strong coupling, when the adiabatic potential in the space of the nuclear displacements that lift the degeneracy has several equivalent minima, tunneling between which leads to a number of peculiar effects. From among the latter, we consider resonant magnetic- and acoustic-wave absorption due to transitions between tunnel and tunnel-rotational levels, the temperature and frequency dependences of the EPR and hyperfine splittings in M\"ossbauer spectra, polarizability, etc. We discuss briefly cases of weak coupling suppression of splitting, due to perturbations that act in the electronic subsystem. For the crystalline states, the interaction of local Jahn--Teller and pseudo-Jahn-Teller distortions leads to structural transitions (the cooperative Jahn--Teller effect). In the case when the local distortions are of the dipole type, the crystal in the ordered phase turns out to be the spontaneously polarized. Relations are obtained between the parameters of the vibronic coupling, the spontaneous polarization, and the Curie temperature for a number of ferroelectrics.