  There are currently many string inspired conjectures about the structure of
the low-energy effective action for super Yang-Mills theories which require
explicit multi-loop calculations. In this paper, we develop a manifestly
covariant derivative expansion of superspace heat kernels and present a scheme
to evaluate multi-loop contributions to the effective action in the framework
of the background field method. The crucial ingredient of the construction is a
detailed analysis of the properties of the parallel displacement propagators
associated with Yang-Mills supermultiples in N-extended superspace.
