Let \((M,g_0)\) be a compact Riemannian manifold of dimension \(n\geq 3\). Denote by \([g_0]\) the set of metrics conformal to \(g_0\). For \(g\in[g_0]\) denote by \(R\), Ric, and \(A_g\) the scalar curvature, the Ricci tensor, and the Schouten tensor, respectively; \(A_g\) is defined by \(2(n- 1)(n- 2)A_g= 2(n- 1)-\text{Ric}_g- R_g-g\). The author studies the regularity and existence of solutions to the equation (1.1) \(f(\lambda)=\varphi\) under the following assumptions: \(\lambda= \lambda(A_g)=(\lambda_1,\dots, \lambda_n)\) denotes the eigenvalues of \(A_g\) with respect to \(g\); \(\varphi\) is a positive, smooth function; \(f\) is a nonlinear function which is defined on an open convex cone \(\Gamma\subset\mathbb{R}^n\) and which is \(>0\) in \(\Gamma\), \(=0\) on \(\partial\Gamma\), concave, invariant under exchange of variables, homogeneous of degree \(\alpha\) for some \(\alpha> 0\), and satisfying the condition \(\partial f/\partial\lambda_i> 0\) \((i= 1,\dots,n)\). For the class (1.1) of fully nonlinear elliptic equations a priori estimates for the interior gradient (Section 4) and the second derivatives (Section 3) are proved. As an application of these estimates the existence of solutions to problem (1.1) is proved if either (Theorem 2.3) the manifold \((M, g_0)\) is locally conformally flat and \(\varphi\equiv 1\), or if (Theorem 2.4) the cone \(\Gamma\) is relatively small, so that any admissible metric \(g\) has positive Ricci curvature. As the author mentions in Remark 2.1, the interior (gradient and second derivatives) estimates have also been proved, independently, by S. Chen (preprint Sept. 2005).