This paper is a continuation of parts I and II [Commun. Pure Appl. Math. 37, 369-402 (1984; Zbl 0598.35047) and 38, 209-252 (1985; Zbl 0598.35048)]. Here is studied the solvability of Dirichlet’s problem in a bounded domain \(\Omega \subset R^ n\) with smooth boundary \(\partial \Omega:\) \[ F(D^ 2u)=\psi \quad in\quad \Omega;\quad u=\phi \quad on\quad \partial \Omega, \] where the function F is defined by a smooth symmetric function \(f(\lambda_ 1,...,\lambda_ n)\) of the eigenvalues \(\lambda =(\lambda_ 1,...,\lambda_ n)\) of the Hessian matrix \(D^ 2u=\{u_{ij}\}\). It is assumed that the equation is elliptic, i.e. \(\partial f/\partial x_ i>0\), for all i, and that f is a concave function.