The present state of theoretical and experimental research on forbidden rotational transitions in polar and nonpolar molecules, initiated mainly in 1971, is reviewed. Three mechanisms whereby forbidden rotational transitions appear are considered: 1) centrifugal distortion, 2) anharmonicity of the potential energy, 3) electron-vibrational interaction (the Jahn-Teller effect). A method is described for calculating the effective dipole moment and the strength of a transition, based on sucessive contact transformations of the Hamiltonian and of the dipole moment operator. Formulas are given for the line strengths of the most important classes of molecules. Results are presented of experimental investigations of centrifugal transitions of tetrahedral hydride molecules by the double resonance and microwave and infrared spectroscopy methods. Particular attention is paid to Jahn-Teller transitions, which have not yet been observed. An approximate estimate is made, for the first time, of the dipole moment of such transitions in triply degenerate electronic states of molecules (such as the ground state of methane) and it is shown that the Jahn-Teller transition strengths can be of the same order as of the anharmonic and centrifugal transitions.