Cohomotopy sets of manifolds (homotopy classes of maps into spheres) can be studied via the Pontryagin-Thom construction. The Pontryagin-Thom construction relates cohomotopy sets to framed submanifolds up to framed cobordism in the domain, which can be computed explicitly in favorable cases. In this talk I recall this theory and will discuss a generalization to mappings between infinite dimensional spaces. This is joint work with Alberto Abbondandolo.