We derive general conditions for 100% frequency conversion in any doubly resonant nonlinear cavity, for both second- and third-harmonic generation via $\chi^(2)$ and $\chi^(3)$ nonlinearities. We find that conversion efficiency is optimized for a certain critical power depending on the cavity parameters, and assuming reasonable parameters we predict 100% conversion using milliwatts of power or less. These results follow from a semi-analytical coupled-mode theory framework which is generalized from previous work to include both $\chi^(2)$ and $\chi^(3)$ media as well as inhomogeneous (fully vectorial) cavities, analyzed in the high-efficiency limit where down-conversion processes lead to a maximum efficiency at the critical power, and which is verified by direct finite-difference time-domain (FDTD) simulations of the nonlinear Maxwell equations. Explicit formulas for the nonlinear coupling coefficients are derived in terms of the linear cavity eigenmodes, which can be used to design and evaluate cavities in arbitrary geometries.