We develop a general theory of duality for Markov processes satisfying Meyer's hypothesis (L) and possessing an excessive reference measure. We make use of a compactification introduced by Walsh which allows a right process and its moderate dual to have strong Markov versions on an enlarged state space. The representation theory for potentials of additive functionals due to Revuz and Sharpe can be extended to this setting. Using this theory, we show that the conatural additive functionals introduced by Garcia-Alvarez are natural additive functionals in the new topology. A general version of Motoo's theorem is given, and the Getoor-Sharpe approach to capacities is extended to this situation. Finally, we show that if the original process satisfies Hunt's hypothesis (H), then a version of the bounded maximum principle holds on the compactified space.