  We present a general framework for automatic continuity results for groups of
isometries of metric spaces. In particular, we prove automatic continuity
property for the group of isometries of the Urysohn space and the Urysohn
sphere, i.e. that any homomorphism from either of these groups into a separable
group is continuous. As a consequence, we get that the group of isometries of
the Urysohn space has unique Polish group topology and the group of isometries
of the Urysohn sphere has unique separable group topology. Moreover, as an
application of our framework we obtain new proofs of the automatic continuity
property for the group $\mathrm{Aut}([0,1],\lambda)$, due to Ben Yaacov,
Berenstein and Melleray and for the unitary group of the infinite-dimensional
separable Hilbert space, due to Tsankov. The results and proofs are stated in
the language of model theory for metric structures.
