An overview of microscopic current-voltage relations applicable to mesoscopic superconductors is presented. These are used to examine a variety of new phenomena, including the change delta G in the two-probe electrical conductance G of a mesoscopic sample due to the switching on of superconductivity. It is predicted that delta G can have an arbitrary, sample dependent sign, have a magnitude much greater than 2e(2)/h and is suppressed by the application of a magnetic field. For an Andreev phase gradiometer formed by attaching a finite width normal wire at 90 degrees to a superconductor, it is predicted that due to quantum interference from an order parameter phase gradient, the conductance of the wire will be an oscillatory function of the supercurrent. For an Andreev interferometer obtained by embedding a pair of superconductors with an order parameter phase difference phi, in a disordered normal host, it is predicted that the phase periodic conductance G(phi) may have a maximum or a minimum at phi = 0. In addition, the amplitude of the ensemble-averaged, 2 pi periodic Fourier component decreases with energy, suggesting the possibility of a cross-over from a 2 pi to pi periodicity with increasing temperature. Finally for a T-shaped normal structure, with a superconducting island located on the vertical leg and a current passing horizontally from left to right, it is predicted that the differential conductance exhibits a slow oscillatory dependence on the position of the superconductor and on the applied voltage.