The classical Pontryagin–Thom isomorphism equates manifold bordism groups with corresponding stable homotopy groups. This construction moreover generalizes to the equivariant context. I will discuss work which establishes a Pontryagin--Thom isomorphism for orbispaces (an orbispace is a "space" which is locally modelled on Y/G for Y a space and G a finite group; examples of orbispaces include orbifolds and moduli spaces of pseudo-holomorphic curves). This involves defining a category of orbispectra and an involution of this category extending Spanier--Whitehead duality. Global homotopy theory also plays a key role.