We present a comprehensive theory of the electrical conductance G of phase-coherent, multi-channel, resonant structures in the presence of superconductivity. When voltages of the order of the level spacing are applied, particle-hole symmetry is broken and our results differ significantly from earlier descriptions. After deriving generalizations of the well-known Breit-Wigner formula, valid in the presence of superconductivity, results for resonant transport in three classes of structure are obtained. First, for a superconducting dot (SDOT) connected to normal contacts (N), we examine the change in conductance as the magnitude of the superconducting order parameter increases from zero. The change is typically negative, except near a normal-state resonance, where large positive changes can occur. Secondly, for a structure comprising a normal (N) contact, a normal dot (NDOT) and a superconducting (S) contact, we predict that finite-voltage, differential conductance resonances are strongly suppressed by the switching on of superconductivity in the S contact. In the weak-coupling limit, resonances which survive have a double-peaked line-shape. Thirdly, analytic results are presented for superconductivity enhanced, quasi-particle interferometers (SEQUINs), which demonstrate that resonant SEQUINs can provide galvanometric magnetic Bur detectors, with a sensitivity in excess of the flux quantum.