  Let G be a vertex transitive graph. A study of the range of simple random
walk on G and of its bridge is proposed. While it is expected that on a graph
of polynomial growth the sizes of the range of the unrestricted random walk and
of its bridge are the same in first order, this is not the case on some larger
graphs such as regular trees. Of particular interest is the case when G is the
Cayley graph of a group. In this case we even study the range of a general
symmetric (not necessarily simple) random walk on G. We hope that the few
examples for which we calculate the first order behavior of the range here will
help to discover some relation between the group structure and the behavior of
the range. Further problems regarding bridges are presented.
