  We introduce a model of the set of all Polish (=separable complete metric)
spaces: the cone $\cal R$ of distance matrices, and consider geometric and
probabilistic problems connected with this object. The notion of the universal
distance matrix is defined and we proved that the set of such matrices is
everywhere dense $G_{\delta}$ set in weak topology in the cone $\cal R$.
  Universality of distance matrix is the necessary and sufficient condition on
the distance matrix of the countable everywhere dense set of so called
universal Urysohn space which he had defined in 1924 in his last paper. This
means that Urysohn space is generic in the set of all Polish spaces. Then we
consider metric spaces with measures (metric triples) and define a complete
invariant: its - matrix distribution. We give an intrinsic characterization of
the set of matrix distributions, and using the ergodic theorem, give a new
proof of Gromov's ``reconstruction theorem'. A natural construction of a wide
class of measures on the cone $\cal R$ is given and for these we show that {\it
with probability one a random Polish space is again the Urysohn space}. There
is a close connection between these questions, metric classification of
measurable functions of several arguments, and classification of the actions of
the infinite symmetric group [V1,V2].
