  This article introduces and investigates the basic features of a dynamical
zeta function for group actions, motivated by the classical dynamical zeta
function of a single transformation. A product formula for the dynamical zeta
function is established that highlights a crucial link between this function
and the zeta function of the acting group. A variety of examples are explored,
with a particular focus on full shifts and closely related variants. Amongst
the examples, it is shown that there are infinitely many non-isomorphic
virtually cyclic groups for which the full shift has a rational zeta function.
In contrast, it is shown that when the acting group has Hirsch length at least
2, a dynamical zeta function with a natural boundary is more typical. The
relevance of the dynamical zeta function in questions of orbit growth is also
considered.
