  For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em
covering number} $N(K , \varepsilon )$ is defined as the smallest number of
balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em
metric entropy}, i.e., the asymptotic behaviour of covering numbers for
(families of) metric spaces is important in many areas of mathematics
(geometry, functional analysis, probability, coding theory, to name a few). In
this paper we give asymptotically correct estimates for covering numbers for a
large class of homogeneous spaces of unitary (or orthogonal) groups with
respect to some natural metrics, most notably the one induced by the operator
norm. This generalizes earlier author's results concerning covering numbers of
Grassmann manifolds; the generalization is motivated by applications to
noncommutative probability and operator algebras. In the process we give a
characterization of geodesics in $U(n)$ (or $SO(m)$) for a class of
non-Riemannian metric structures.
