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In this paper the authors study compact homogeneous spaces $G/K$ (of a Lie group $G$) on which there if defined a $G$-invariant symplectic form $\Omega$. It is an important feature of the paper that very little is assumed concerning $G$ and $K$. The essential assumptions are: (1) $G$ is connected and (2) $K$ is uniform (i.e., $G/K$ is compact). Further, for convenience only and with no loss of generality, it is supposed that $G$ is simply connected and $K$ contains no connected normal subgroup of $G$, i.e., that $G$ acts almost effectively...