  We analyze various supersymmetry multiplets containing the supercurrent and
the energy-momentum tensor. The most widely known such multiplet, the
Ferrara-Zumino (FZ) multiplet, is not always well-defined. This can happen once
Fayet-Iliopoulos (FI) terms are present or when the Kahler form of the target
space is not exact. We present a new multiplet S which always exists. This
understanding of the supersymmetry current allows us to obtain new results
about the possible IR behavior of supersymmetric theories. Next, we discuss the
coupling of rigid supersymmetric theories to supergravity. When the theory has
an FZ-multiplet or it has a global R-symmetry the standard formalism can be
used. But when this is not the case such simple gauging is impossible. Then, we
must gauge the current S. The resulting theory has, in addition to the graviton
and the gravitino, another massless chiral superfield Phi which is essential
for the consistency of the theory. Some of the moduli of various string models
play the role of Phi. Our general considerations, which are based on the
consistency of supergravity, show that such moduli cannot be easily lifted thus
leading to constraints on gravity/string models.
