  We introduce the concept of action space, a set $\boldsymbol{X}$ endowed with
an action cost $\mathsf{a}:(0,+\infty)\times \boldsymbol{X}\times
\boldsymbol{X}\to [0,+\infty)$ satisfying suitable axioms, which turn out to
provide a `dynamic' generalization of the classical notion of metric space.
Action costs naturally arise as dissipation terms featuring in the Minimizing
Movement scheme for gradient flows, which can then be settled in general action
spaces.
  As in the case of metric spaces, we will show that action costs induce an
intrinsic topological and metric structure on $\boldsymbol{X}$. Moreover, we
introduce the related action functional on paths in $\boldsymbol{X}$,
investigate the properties of curves of finite action, and discuss their
absolute continuity. Finally, under a condition akin to the approximate
mid-point property for metric spaces, we provide a dynamic interpretation of
action costs.
