Loop manifolds (or free loop spaces) are the infinite dimensional spaces of free loops on a manifold. They play an important role in the development of elliptic genera and elliptic cohomology and have found interesting applications in Quantum Field Theories and quantum groups.

In this talk I will describe the successive notions of orientation, spin and string structures on a compact manifold and in particular how the latter is related to objects on the loop manifold. There remain serious difficulties in doing differential analysis on these infinite dimensional spaces and I will try to show where the problems lie and some possible paths forward as illustrated by ongoing work with Chris Kottke.