The author considers fully nonlinear nonvariational elliptic systems of the type \[ a(H(u))=f \quad \text{in }\Omega \subset \mathbb{R}^ n, \tag{1} \] where \(u:\Omega \to \mathbb{R}^ N\) is vector valued, and \(H(u):=\{D_ iD_ ju\}\) \((i,j=1,\dots,n)\) is the matrix of the second partial derivatives of \(u\). The function \(a\) satisfies a ``pseudo-monotonicity condition'' which is weaker than the usual ellipticity condition for differentiable \(a\). Let \(u \in H^ 2 (\Omega)\) be a solution of (1) with \(f=0\). Then the so called fundamental interior estimates for \(H(u)\), \(Du:=(D_ 1u,\dots,D_ nu)\) and \(u\) are satisfied, and \(u \in H^{2,q}_{loc} (\Omega)\) for some \(q>2\). Moreover, let \(u \in H^ 2(\Omega)\) be a solution of (1) with \(f \in {\mathcal L}^{2,\mu}(\Omega)\) and \(n<\mu<2+n(1-{2 \over q})\). Then \(H(u)\) is Hölder continuous. Finally, the author proves a partial Hölder continuity result for solutions of more general systems of the type \(a(x,u,Du,H(u))=b(x,u,Du)\).