The problem \(F(D^2u)\equiv f(\lambda[D^2u])= \psi\) with Dirichlet boundary conditions is studied. Here the problem is considered on a domain in \(\mathbb{R}^n\) and \(f\) is either a symmetric function \(S_k(\lambda)\) on \(\mathbb{R}^n\) or a quotient of elementary symmetric functions \(S_{k,l}(\lambda)= S_k(\lambda)/S_l(\lambda)\) with \(n\geq k>l>1\), \(\lambda\) denotes the eigenvalues \(\lambda_1,\lambda_2,\dots, \lambda_n\) of the Hessian matrix \(D^2u\) and \(\psi\) is a given function. Existence and uniqueness results extending a previous work by Caffarelli-Nirenberg-Spruck are obtained. The first main result is Theorem 1.1, stating that the above problem with the boundary condition \(u=\phi\) on \(\partial\Omega\), \(\Omega\) a bounded uniformly \((k-1)\)-convex domain in \(\mathbb{R}^n\), with \(\partial\Omega\in C^{3,1}\), \(\phi\in C^{3,1}(\partial\Omega)\) and \(\psi\in C^{1,1}(\overline\Omega)\) positive, has a unique solution for admissible \(u\in C^{3,\alpha}(\overline\Omega)\) for any \(0<\alpha<1\). Then a similar result is proved for \(f\) satisfying suitable assumptions provided the curvatures of \(\partial\Omega\), \(\kappa_1,\dots,\kappa_{n-1}\) are such that \((\kappa_1,\dots,\kappa_{n-1},R)\in\Gamma\) for some open convex symmetric cone \(\Gamma\) in \(\mathbb{R}^n\) with vertex at the origin. The geometric conditions on \(\Omega\) are necessary for constant boundary conditions. The main point in the proof is a new technique for obtaining estimates for the double normal second derivatives. This is done in Section 2. A new shorter proof for some known results is given in Section 3 by using the same arguments. Some extensions (degenerate problems, general domains, curvature equations) can be found at the end of the paper.