In the first part of the talk, I will review recent advances that allow to define supersymmetric field theories on rather generic compact manifolds, with a special focus on N=1 theories in four dimensions. I will then explain how one can obtain general constrains on the parameter dependence of supersymmetric partition functions on those compact spaces. For N=1 theories, we need to consider compactifications on Hermitian four-manifolds in order to preserve one supercharge. Then, one can show that the partition function is actually independent of the Hermitian metric, while it depends holomorphically on the complex structure moduli of the underlying complex four-manifold.
If times permits, I will also discuss a very interesting parallel story for 3d N=2 theories on three-manifolds.
