  Given a topological group $G$ and a unitary representation $U$ of $G$, we
consider the problem of classifying the positive operator measures which are
based on a $G$-homogeneous space $X$ and covariant with respect to the
representation $U$. We completely solve the problem in two cases: 1) when $G$
is abelian; 2) when $G$ is not abelian, $X$ has compact stbiliser in $G$ and
the representation $U$ is irreducible. We then give an application of the above
results to the characterisation of covariant position and momentum observables
in quantum mechanics, and to the determination of those covariant position and
momentum observables that are jointly measurable.
