Summary: We study optimization problems involving eigenvalues of symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not differentiable at those points where they coalesce. We present a general framework for a smooth (differentiable) approach to such problems. It is based on the concept of transversality borrowed from differential geometry. In that framework we discuss first- and second-order optimality conditions and rates of convergence of the corresponding second-order algorithms. Finally, we present some results on the sensitivity analysis of such problems.