It is now possible to do quantum reactive scattering calculations on the state-to-state dynamics of gas phase chemical reactions involving four atoms. This article gives an overview of this recent progress. A survey is also given of results obtained using the rotating bond approximation (RBA). This general method is state-selective in the stretching and bending vibrations of ABC, the vibrations of AB and CD, and the rotational states of AB in the reaction ABC + D --> AB + CD. Results for the OH + H-2, OH + CO, OH + HCl, and OH + HBr reactions are presented and compared with experiment. Calculations of product rotational and vibrational distributions, differential cross sections, bond and mode-selective effects, lifetimes of collision complexes, and unusual temperature dependences in rate constants are discussed. The predictions and comparisons with experiment show that the RBA is a useful and versatile quantum theory for studying the state-to-state dynamics of reactions involving polyatomic molecules.