The Car-Parrinello (CP) method made molecular dynamics simulation with on-the-fly computation of interaction potentials from electronic structure theory computationally feasible. The method reformulates ab initio molecular dynamics (AIMD) as a two-component classical dynamical system. This approach proved to be valuable far beyond the original CP molecular dynamics method. The modern formulation of Born-Oppenheimer (BO) dynamics is based on the same basic principles and can be derived from the same Lagrange function as the CP method. These time-reversible BO molecular dynamics methods allow higher accuracy and efficiency while providing similar longtime stability as the CP method. AIMD is used in many fields of computational physics and chemistry. Its applications are instrumental in fields as divers as enzymatic catalysis and the study of the interior of planets. With its versatility and predictive power, AIMD has become a major approach in atomistic simulations. (C) 2011 John Wiley & Sons, Ltd.