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There are currently many string inspired conjectures about the structure ofthe low-energy effective action for super Yang-Mills theories which requireexplicit multi-loop calculations. In this paper, we develop a manifestlycovariant derivative expansion of superspace heat kernels and present a schemeto evaluate multi-loop contributions to the effective action in the frameworkof the background field method. The crucial ingredient of the construction is adetailed analysis of the properties of the parallel displacement propagatorsassociated with Yang-Mills supermultiples in N-extended superspace.Comment: 32 pages, latex, 7 EPS figures. v2: references, comments added, typos  corrected, incorrect `skeleton' conjecture in sect. 3 replaced by a more  careful treatment. v3: typos corrected, final version published in JHE

On the background field method beyond one loop: a manifestly covariant derivative expansion in super Yang-Mills theories