  In this review, we give a pedagogical introduction to a systematic framework
for constructing and analyzing supersymmetric field theories on curved
spacetime manifolds. The framework is based on the use of off-shell
supergravity background fields. We present the general principles, which
broadly apply to theories with different amounts of supersymmetry in diverse
dimensions, as well as specific applications to N=1 theories in four dimensions
and their three-dimensional cousins with N=2 supersymmetry.
