  WWe define the notion of a random metric space and prove that with
probability one such a space is isometricto the Urysohn universal metric space.
The main technique is the study of universal and random distance matrices; we
relate the properties of metric (in particulary universal) space to the
properties of distance matrices. We show the link between those questions and
classification of the Polish spaces with measure (Gromov or metric triples) and
with the problem about S_{\infty}-invariant measures in the space of symmetric
matrices. One of the new effects -exsitence in Urysohn space so called
anarchical uniformly distributed sequences. We give examples of other
categories in which the randomness and universality coincide (graph, etc.).
