  We present a systematic classification of counterterms of four-dimensional
supersymmetric field theories on curved space, obtained as the rigid limit of
new minimal supergravity. These are supergravity invariants constructed using
the field theory background fields. We demonstrate that if the background
preserves two supercharges of opposite chirality, then all dimensionless
counterterms vanish, implying that in this case the supersymmetric partition
function is free of ambiguities. When only one Euclidean supercharge is
preserved, we describe the ambiguities that appear in the partition function,
in particular in the dependence on marginal couplings.
