Phase-coherent diffusive transport through mesoscopic hybrid superconductor-normal-metal tunneling structures is investigated. For a N-s-S two-barrier tunneling system with bulk S and N electrodes coupled by a mesoscopic superconducting constriction s, zero-bias conductance and nonlinear I-V curves are calculated under the assumption that the dwell times of quasiparticles in the s region are shorter than the inelastic relaxation time. It is shown that the low voltage conductance of this system determined by the Andreev reflection processes may exceed the conductance in the normal state and its value is very sensitive to the weak pairing interaction of electrons in the s region. We show that even weak pairing electron interaction may result in the significant qualitative and quantitative change of the conductance temperature dependence with respect to the case of structures with the normal mesoscopic region. We calculate the I-V curves and show that they depend on the applied voltage in a nonmonotonic way, therefore differential conductance becomes negative with increasing voltage. Such behavior is due to the voltage dependence of the order parameter in the constriction and the phase difference cp between the S and s superconductors. It is shown that if the tunneling processes determine the form of the quasiparticle distribution function in the s superconductor, the phase cp is stationary at arbitrary voltages. For quasiparticle tunneling interferometers in which the mesoscopic superconductor s couples the superconductor S and the normal metal N, the zero-bias conductance, as a function of the phase difference between the S electrodes is investigated. It is shown that the amplitude of the conductance oscillations may exceed the conductance of this structure in the normal state. [S0163-1829(99)07329-4].