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We develop the regularity theory of viscosity solutions to transmissionproblems for fully nonlinear second order uniformly elliptic equations. Ourresults give a complete theory of existence, uniqueness, comparison principle,and regularity of solutions to flat interface transmission problems; and the$C^{0,\alpha}$, $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity of viscositysolutions up to the transmission surface for the case of curved interfaces.

Regularity of viscosity solutions to fully nonlinear elliptic transmission problems