  Following results of Bourgain and Gorelik we show that the spaces $\ell_p$,
$1<p<\infty$, as well as some related spaces have the following uniqueness
property: If $X$ is a Banach space uniformly homeomorphic to one of these
spaces then it is linearly isomorphic to the same space. We also prove that if
a $C(K)$ space is uniformly homeomorphic to $c_0$, then it is isomorphic to
$c_0$. We show also that there are Banach spaces which are uniformly
homeomorphic to exactly $2$ isomorphically distinct spaces.
