  We present a uniform treatment of rigid supersymmetric field theories in a
curved spacetime $\mathcal{M}$, focusing on four-dimensional theories with four
supercharges. Our discussion is significantly simpler than earlier treatments,
because we use classical background values of the auxiliary fields in the
supergravity multiplet. We demonstrate our procedure using several examples.
For $\mathcal{M}=AdS_4$ we reproduce the known results in the literature. A
supersymmetric Lagrangian for $\mathcal{M}=\mathbb{S}^4$ exists, but unless the
field theory is conformal, it is not reflection positive. We derive the
Lagrangian for $\mathcal{M}=\mathbb{S}^3\times \mathbb{R}$ and note that the
time direction $\mathbb{R}$ can be rotated to Euclidean signature and be
compactified to $\mathbb{S}^1$ only when the theory has a continuous
R-symmetry. The partition function on $\mathcal{M}=\mathbb{S}^3\times
\mathbb{S}^1$ is independent of the parameters of the flat space theory and
depends holomorphically on some complex background gauge fields. We also
consider R-invariant $\mathcal{N}=2$ theories on $\mathbb{S}^3$ and clarify a
few points about them.
