  We construct supersymmetric field theories on Riemannian three-manifolds M,
focusing on N=2 theories with a U(1)_R symmetry. Our approach is based on the
rigid limit of new minimal supergravity in three dimensions, which couples to
the flat-space supermultiplet containing the R-current and the energy-momentum
tensor. The field theory on M possesses a single supercharge, if and only if M
admits an almost contact metric structure that satisfies a certain
integrability condition. This may lead to global restrictions on M, even though
we can always construct one supercharge on any given patch. We also analyze the
conditions for the presence of additional supercharges. In particular, two
supercharges of opposite R-charge exist on every Seifert manifold. We present
general supersymmetric Lagrangians on M and discuss their flat-space limit,
which can be analyzed using the R-current supermultiplet. As an application, we
show how the flat-space two-point function of the energy-momentum tensor in N=2
superconformal theories can be calculated using localization on a squashed
sphere.
