In the last few decades, billiard systems have provided a most appropriate model for the understanding and discussion of classical chaotic Hamiltonian systems and their quantum counterparts. In two dimensions the Schrödinger equation for quantum billiards coincides with the Helmholtz equation for flat cylindrical microwave cavities of corresponding shape. This correspondence provides the basis for experimental studies of quantum manifestations of classical chaos. We present results of experiments with normal and superconducting microwave cavities, where the former are used for the measurement of wavefunctions, the latter for the measurement of energy eigenvalues.