  Computer simulation methods, such as Monte Carlo or Molecular Dynamics, are
very powerful computational techniques that provide detailed and essentially
exact information on classical many-body problems. With the advent of ab-initio
molecular dynamics, where the forces are computed on-the-fly by accurate
electronic structure calculations, the scope of either method has been greatly
extended. This new approach, which unifies Newton's and Schr\"odinger's
equations, allows for complex simulations without relying on any adjustable
parameter. This review is intended to outline the basic principles as well as a
survey of the field. Beginning with the derivation of Born-Oppenheimer
molecular dynamics, the Car-Parrinello method and the recently devised
efficient and accurate Car-Parrinello-like approach to Born-Oppenheimer
molecular dynamics, which unifies best of both schemes are discussed. The
predictive power of this novel second-generation Car-Parrinello approach is
demonstrated by a series of applications ranging from liquid metals, to
semiconductors and water. This development allows for ab-initio molecular
dynamics simulations on much larger length and time scales than previously
thought feasible.
