  We study isometric $G$-spaces and the question of when their maximal
equivariant compactification is the Gromov compactification (meaning that it
coincides with the compactification generated by the distance functions to
points). Answering questions of Pestov, we show that this is the case for the
Urysohn sphere and related spaces, but not for the unit sphere of the Gurarij
space.
  We show that the maximal equivariant compactification of a separably
categorical metric structure $M$ under the action of its automorphism group can
be identified with the space $S_1(M)$ of 1-types over $M$, and is in particular
metrizable. This provides a unified understanding of the previous and other
examples. In particular, the maximal equivariant compactifications of the
spheres of the Gurarij space and of the $L^p$ spaces are metrizable.
  We also prove a uniform version of Effros' Theorem for isometric actions of
Roelcke precompact Polish groups.
