  The paper is devoted to the study of the Gromov-Hausdorff proper class,
consisting of all metric spaces considered up to isometry. In this class, a
generalized Gromov-Hausdorff pseudometric is introduced and the geometry of the
resulting space is investigated. The first main result is a proof of the
completeness of the space, i.e., that all fundamental sequences converge in it.
Then we partition the space into maximal proper subclasses consisting of spaces
at a finite distance from each other. We call such subclasses clouds. A
multiplicative similarity group operates on clouds, multiplying all the
distances of each metric space by some positive number. We present examples of
similarity mappings transferring some clouds into another ones. We also show
that if a cloud contains a space that remains at zero distance from itself
under action of all similarities, then such a cloud contracted to this space.
In the final part, we investigate subsets of the real line with respect to
their behavior under various similarities.
