We survey geometric constructions of characteristic classes associated to certain infinite rank bundles on the loop space
of a manifold
. 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
, as the curvature forms take values in pseudodifferential operators. The leading order classes lead to a restatement of the
-index theorem on
, provide generators for the cohomology of loop groups, and for Maps(
) are related to Gromov-Witten invariants. The Wodzicki classes have applications to the topology of diffeomorphism groups of certain circle bundles over Kaehler surfaces.