This review is about potential energy surfaces. In general, the potential energy, V, of a system of N interacting particles, is a function of 3N spatial coordinates: V=V({r_i}), (i = 1 -> N), or V=V(X), where X is the 3N-dimensional configuration vector. The potential energy surface (PES) is therefore a 3N-dimensional object embedded in a 3N+1 dimensional space, where the extra dimension corresponds to the value of the potential energy function.

In the present work we will be concerned only with the ground state PES in the Born-Oppenheimer approximation, although extensions to excited electronic states are certainly possible in principle. We will also employ classical mechanics and neglect quantum effects, which are expected to be small for the systems of interest here. The PES then determines, either directly or indirectly, the structure, dynamics and thermodynamics of the system. Mechanically stable configurations correspond to local minima of V, while the gradient of V tells us the forces on the various atoms. Thermodynamic properties also depend upon V via ensemble averages. Even systems composed of hard spheres or discs have a potential energy surface, albeit a rather strange one, since the energy is either zero or infinity. For the more realistic potentials considered in this review it is possible to provide further insight into dynamics and thermodynamics by considering the densities of states associated with
local minima and elementary events consisting of transitions between minima. The theory and validity of this approach is considered later.

For many small molecules relatively accurate quantum mechanical calculations provide a useful global picture of the PES. However, here we are interested in treating systems with far more degrees of freedom, where it is often necessary to find different approaches. Two areas of current research have stimulated our interest, namely protein folding and the properties of glasses. Understanding the behaviour of such systems at a fundamental level, in terms of the underlying PES, has motivated us to study larger systems. Of course, much thought has already been given to the question of how dynamics and thermodynamics emerges from the underlying PES. However, it is only relatively recently that increased computer power and improved algorithms have combined to make detailed studies of realistic systems possible. These advances apply particularly to the characterization of pathways between minima and hence the connectivity of the PES, which is critical in determining relaxation dynamics. It is not enough simply to characterize a large sample of minima or investigate the effects of a particular barrier height distribution.