  We solve the Dirichlet problem for fully nonlinear elliptic equations on
Riemannian manifolds under essentially optimal structure conditions, especially
with no restrictions to the curvature of the underlying manifold and the second
fundamental form of its boundary. The main result (Theorem 1.1) includes a new
(and optimal) result in the Euclidean case. We introduce some new ideas and
methods in deriving a priori estimates, which can be used to treat other types
of fully nonlinear elliptic and parabolic equations on real or complex
manifolds.
