  We address the problem of the construction of quantum walks on Cayley graphs.
Our main motivation is the relationship between quantum algorithms and quantum
walks. In particular, we discuss the choice of the dimension of the local
Hilbert space and consider various classes of graphs on which the structure of
quantum walks may differ. We completely characterise quantum walks on free
groups and present partial results on more general cases. Some examples are
given, including a family of quantum walks on the hypercube involving a
Clifford Algebra.
