We present a variational method for computing the optimal aerodynamic performance of conventional and compound helicopters in trimmed flight. The optimal circulation distribution minimizes the sum of the induced and viscous power required to develop a prescribed lift and/or thrust, subject to any constraints that the helicopter be trimmed in pitch and roll. The minimum total power circulation distribution problem is cast as a variational problem, which in turn is solved efficiently using a vortex-lattice technique. Included in the analysis is the viscous profile power, which is estimated at each airfoil section using an experimental or numerically computed drag polar. The resulting analysis-which is the viscous helicopter analogue of Goldstein's inviscid propeller theory-gives rigorous upper bounds on the performance of conventional and compound helicopters and may be used to predict the rotor/wing loadings that produce optimal performance. We show that helicopters with either coaxial rotors or a wing and rotor in combination can substantially reduce power loss by producing a more efficient wake structure and by reducing the induced power associated with roll trim. © 2010 The American Helicopter Society.