We present our theoretical results obtained from the random-matrix approach to quantum chaotic scattering for systems with partially violated time-reversal (T) invariance. They were tested most precisely in measurements of scattering amplitudes of a chaotic microwave resonator in the regime of isolated and overlapping resonances. Below a certain excitation frequency the resonator simulates a quantum billiard, whose eigenvalues manifest themselves in the spectra as resonances with average spacing d and width Γ. Violation of T invariance is achieved with a magnetized ferrite inside the cavity. The experimental observables are complex scattering (S)-matrix elements, measured for the resonator with and without T invariance as a function of frequency. Particular emphasis is given to S-matrix correlation functions in the regime of weakly overlapping resonances, i.e., Γ /d < 1, and their comparison to results from the theory of chaotic scattering developed in nuclear reaction theory. We also present results on the distribution of the S-matrix elements and higher order correlation functions. Here, a focus is the transition from the regime of weakly overlapping resonances to the Ericson regime, i.e., from non-exponential to exponential decay of the system of resonances.