  We study the logical properties of infinite geometric random graphs,
introduced by Bonato and Janssen. These are graphs whose vertex set is a dense
``generic'' subset of a metric space, where two vertices are adjacent with
probability $p>0$ provided the distance between them is bounded by some
constant number. We prove that for a large class of metric spaces, including
circles, spheres and the complete Urysohn space, almost all geometric random
graphs on a given space are elementary equivalent. Moreover, their first-order
theory can reveal geometric properties of the underlying metric space.
