  In this paper we review and streamline some results of Kirillov, Olshanski
and Pickrell on unitary representations of the unitary group $\U(\cH)$ of a
real, complex or quaternionic separable Hilbert space and the subgroup
$\U_\infty(\cH)$, consisting of those unitary operators $g$ for which $g - \1$
is compact. The Kirillov--Olshanski theorem on the continuous unitary
representations of the identity component $\U_\infty(\cH)_0$ asserts that they
are direct sums of irreducible ones which can be realized in finite tensor
products of a suitable complex Hilbert space. This is proved and generalized to
inseparable spaces. These results are carried over to the full unitary group by
Pickrell's Theorem, asserting that the separable unitary representations of
$\U(\cH)$, for a separable Hilbert space $\cH$, are uniquely determined by
their restriction to $\U_\infty(\cH)_0$. For the $10$ classical infinite rank
symmetric pairs $(G,K)$ of non-unitary type, such as $(\GL(\cH),\U(\cH))$, we
also show that all separable unitary representations are trivial.
