Summary: A method is developed for placing bounds on the electrical contact conductance between contacting elastic bodies with rough surfaces. An analogy is first established between contact conductance and the incremental stiffness in the mechanical contact problem. Results from contact mechanics and the reciprocal theorem are then used to bracket the mechanical load-displacement curve between those for two related smooth contact problems. This enable bounds to be placed on the incremental stiffness, and hence on the electrical conductance for the rough contact problem. The method is illustrated by two simple examples, but its greatest potential probably lies in establishing the maximum effect of neglected microscales of roughness in a solution of the contact problem for bodies with multiscale or fractal roughness.