  We consider the structure of effective lagrangians describing the low-energy
dynamics of supersymmetric theories in which a global symmetry $G$ is
spontaneously broken to a subgroup $H$ while supersymmetry is unbroken. In
accordance with the supersymmetric Goldstone theorem, these lagrangians contain
Nambu--Goldstone superfields associated with a coset space $G^c / \hat{H}$,
where $G^c$ is the complexification of $G$ and $\hat{H}$ is the largest
subgroup of $G^c$ that leaves the order parameter invariant. The lagrangian may
also contain additional light matter fields. To analyze the effective
lagrangian for the matter fields, we first consider the case where the
effective lagrangian is obtained by integrating out heavy modes at weak
coupling (but including non-perturbative effects such as instantons). We show
that the superpotential of the matter fields is $\hat{H}$ invariant, which can
give rise to non-trivial relations among independent $H$-invariants in the
superpotential. We also show that the Kahler potential of the matter fields can
be restricted by a remnant of $\hat{H}$ symmetry. These results are
non-perturbative and have a simple group-theoretic interpretation. When we
relax the weak-coupling constraint, there appear to be additional possibilities
for the action of $\hat{H}$ on the matter fields, hinting that the constraints
imposed by $\hat{H}$ may be even richer in strongly coupled theories.
