Old and new problems of the foundations of quantum mechanics are viewed from the new perspective provided by a generalization of the mathematical formalism encompassing positive operator-valued measures. One objective is to demonstrate the crucial role the generalized formalism plays in fundamental issues as well as in practical applications, and to contribute to the development of the operational approach. A second objective is the development of an empiricist interpretation of this approach, duly taking into account the role played by the measuring instrument in quantum mechanical measurements. Copenhagen and anti-Copenhagen interpretations are critically assessed, and found to be wanting due to insufficiently taking into account the measurement interaction. The Einstein-Podolsky-Rosen problem and the problem of the Bell inequalities are discussed, starting from this new perspective. An explanation of violation of the Bell inequalities is developed, providing an alternative to the usual explanation on the basis of non-locality. This treatise is based on lecture notes of an advanced course on the foundations of quantum mechanics.