How to compute the volume in high dimension?

In some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones. Approximating the volume of a convex n-dimensional body, given by an oracle is one of the areas where this difference can be proved. In general, if we use a deterministic algorithm to approximate the volume, it requires exponentially many oracle questions in terms of n as n[arrow right]∞. Dyer, Frieze and Kannan gave a randomized polynomial approximation algorithm for the volume of a convex body K⊆ Rn, given by a membership oracle. The DKF algorithm was improved in a sequence of papers. The area is full of deep and interesting problems and results. This paper is an introduction to this field and also a survey.