We classify the
primitive distance-transitive representations of the Fischer sporadic
simple groups and their automorphism groups. It turns out
that the only primitive distance-transitive representations of these groups
are their rank 3 representations. In the process of our work,
we also classify and study the primitive multiplicity-free
permutation representations of these Fischer groups.
Our methods, which we describe in some detail,
demonstrate the use of computational
and randomized techniques in the classification of distance-transitive
graphs and the study of very large permutation representations.