Commutative homogeneous spaces and co-isotropic symplectic actions

The paper is a survey of relationships among the following possible properties of a Riemannian homogeneous space $X=G/K$: Selberg's property of weak symmetry, commutativity of the algebra of $K$-invariant measures on $X$, commutativity of the algebra of $G$-invariant differential operators on $X$, commutativity of the Poisson algebra of $G$-invariant functions on the cotangent bundle of the space $X$, and (if $G$ is a reductive group) the property of the spectrum of the linear representation of the group $G$ on the algebra of polynomial functions on $X$ being multiplicity-free. Diverse results on structure and classification are presented, including the author's classification of irreducible Riemannian homogeneous spaces of Heisenberg type for which the Poisson algebra of invariant functions on the cotangent bundle is commutative.