In this talk I will discuss supersymmetric field theories on Euclidean five-manifolds. The talk will focus on theories with less than maximal supersymmetry and the interplay between supersymmetry, geometry and supersymmetric observables. A central role will be taken by the generalization of the notion of an almost complex structure to odd dimensions. This appears naturally in the presence of supersymmetry. If it is integrable in a sense to be specified, it yields an interesting class of geometries for which contributions to the partition function simplify considerably.