In harmonically excited vibrating systems one can observe for certain exciting frequencies a strong growth of the vibration amplitudes. This phenomenon is usually called resonance and it is characteristic for undamped and lightly damped vibrating systems. For multi-dof-systems there may exist not only one but possibly more resonance frequencies. Since the occurance of resonance in vibrating systems is undesirable, due to the large amplitudes, the phenomenon should be more closely investigated. This investigation can be limited to stable vibrating systems, since for an unstable system the natural vibrations are already unbounded so that these systems are of no interest for engineering applications. The formulation of the conditions for resonance will be preceded by a discussion of the elementary frequency response matrix based on the idea of a magnification function for the principal or normal vibrations. This formulation implies necessary conditions for strict resonance and conditions for resonance phenomena. In multi-dof-vibrating systems there may also appear pseudoresonances with finite amplitudes beside strict resonances with infinite amplitudes. Conditions for both cases will be derived. Moreover, one can find excitation vectors which certainly lead to pseudoresonance. Also, in multi-dof-vibrating systems the amplitudes for individual state variables may vanish for certain excitation frequencies.