Traces and characteristic classes in infinite dimensions

We survey geometric constructions of characteristic classes associated to certain infinite rank bundles on the loop space LM of a manifold M. There are two types of classes, which arise from applying either the leading order trace or the Wodzicki residue to the curvature of natural connections on TLM, as the curvature forms take values in pseudodifferential operators. The leading order classes lead to a restatement of the S1-index theorem on LM, provide generators for the cohomology of loop groups, and for Maps(S²,M) are related to Gromov-Witten invariants. The Wodzicki classes have applications to the topology of diffeomorphism groups of certain circle bundles over Kaehler surfaces.