This paper reviewsab initioinvestigations of the Renner-Teller effect in a number of triatomic molecules. The potential surfaces for nuclear motion are calculated using the MRD-CI method. The calculation of vibronic energy levels and wave functions is performed using a simple matrix method for solution of the corresponding Schrodinger equation. In this approach the potential and the kinetic energy operators are expanded in polynomial series in the bending coordinate and the hamiltonian is diagonalized in a basis consisting of products of the electronic wave functions (or their linear combinations) calculated in the Born-Oppenheimer approximation with eigenfunctions of a two-dimensional harmonic oscillator. The method allows for the treatment of large-amplitude bending vibrations and an accurate consideration of non-adiabatic effects. In the second part of this paper the effect of various approximations, as well as some interesting technical details of calculations are discussed. In the third and fourth parts of this review, a summary of results of various calculations is given. A systematic study of dihydrides of the atoms belonging to the first two rows of the Periodic Table allows the recognition of some general trends concerning the equilibrium structure and shapes of the corresponding potential surfaces. A comparison with available experimental data shows that the results ofab initiocalculations permit a reliable representation of observed spectra. Further, the structure of yet unknown spectra can be predicted.