In this talk we propose a method to estimate the density matrix \rho of a d-level quantum system by measurements on the N-fold system. The scheme is based on covariant observables and representation theory of unitary groups and it extends previous results concerning the estimation of the spectrum of \rho. We show that it is consistent (i.e. the original input state is recovered with certainty for N \to infinity) and analyze its large deviation behavior. In addition we calculate explicitly the corresponding rate function which describes the exponential decrease of error probabilities in the limit of infinitely many input systems. For pure input states, or if \rho is mixed but only information about its spectrum is required, we then show that the proposed scheme is optimal in the sense that it provides the fastest possible decay of error probabilities. In the general case, however, the optimality question remains open.