  A quantum walk, \emph{i.e.}, the quantum evolution of a particle on a graph,
is termed \emph{scalar} if the internal space of the moving particle (often
called the coin) has a dimension one. Here, we study the existence of scalar
quantum walks on Cayley graphs, which are built from the generators of a group.
After deriving a necessary condition on these generators for the existence of a
scalar quantum walk, we present a general method to express the evolution
operator of the walk, assuming homogeneity of the evolution. We use this
necessary condition and the subsequent constructive method to investigate the
existence of scalar quantum walks on Cayley graphs of various groups presented
with two or three generators. In this restricted framework, we classify all
groups -- in terms of relations between their generators -- that admit scalar
quantum walks, and we also derive the form of the most general evolution
operator. Finally, we point out some interesting special cases, and extend our
study to a few examples of Cayley graphs built with more than three generators.
