Gauss' continued fraction map has been a prototypical example in the study of dynamical systems, but few of its properties generalize to higher dimensions. A typical problem is to simultaneously approximate a set of irrationals by rationals with common denominator. For such a problem, computer experiments have been very useful and I will describe some recent developments in this area:

1. Properties of the sequence of best-approximation denominators
2. Furtwängler's algorithm
3. Klein polyhedra
4. Computer construction of the worst approximable pair
5. Statistical properties of the continued fraction of a cubic irrational

A major aim of the talk will be to illustrate the interplay between dynamical systems, number theory, and computing.