  We develop the regularity theory of viscosity solutions to transmission
problems for fully nonlinear second order uniformly elliptic equations. Our
results 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 viscosity
solutions up to the transmission surface for the case of curved interfaces.
