In this talk we consider optimization of polynomials in non-commuting variables. More precisely, we are interested in maximal or minimal eigenvalues or traces a polynomial can reach by being evaluated at symmetric or self-adjoint matrices of norm at most 1, or even being evaluated at bounded operators. Those problems are of interest e.g. in Quantum Mechanics as shown by Pironio, Navscués and Acín, as well as in Quantum Statistics. Like the Lasserre relaxation in the case of polynomial optimization in commuting variables, our optimization problem can be relaxed by using sum of squares equivalents for NC polynomials or more general by representations lying in an appropriate quadratic module; both resulting in an SDP. We will present the relaxation scheme and present the current knowledge on the question of convergence of this relaxation scheme in the matrix and in the operator case.