  On the world-volume of an $M$-theory five-brane propagates a two-form with
self-dual field strength. As this field is non-Lagrangian, there is no obvious
framework for determining its partition function. An analogous problem exists
in Type IIB superstring theory for the self-dual five-form. The resolution of
these problems and definition of the partition function is explained. A more
complete analysis of perturbative anomaly cancellation for $M$-theory
five-branes is also presented, uncovering some surprising details.
