This paper deals with the classical Bernoulli free boundary problem. We are interested in solving some shape optimization problems related to this free boundary problem. We prove the continuous dependence of the solution with respect to the data K, working with Hausdorff convergence. We can deduce an existence result for a large class of shape optimization problems. Finally, we give some ideas for a numerical method, based on the use of conformal mappings, to solve such problems in two dimensions.