  The class A of countable groups that admit a faithful, transitive, amenable
-- in the sense that there is an invariant mean -- action on a set has been
widely investigated in the past. In this paper, we no longer require the action
to be transitive, but we ask for it to preserve a locally finite metric (and
still to be faithful and amenable). The groups having such actions are those
that embed into a totally disconnected amenable locally compact group. Then we
focus on the subclass A 1 of groups for which the actions are moreover
transitive. This class is strictly contained into A and includes non-amenable
groups. An important particular case of actions preserving a locally finite
metric is given by actions by automorphisms of locally finite connected graphs.
We take this opportunity, in our partly expository paper, to review some nice
results about amenable actions in this setting.
