  The theory of superconductivity in thin SN sandwiches (bilayers) in the
diffusive limit is developed within the standard Usadel equation method, with
particular emphasis on the case of very thin superconductive layers, d_S <<
d_N. The proximity effect in the system is governed by the interlayer interface
resistance (per channel) \rho_{int}. The case of relatively low resistance
(which can still have large absolute values) can be completely studied
analytically. The theory describing the bilayer in this limit is of BCS type
but with the minigap (in the single-particle density of states) E_g << \Delta
substituting the order parameter \Delta in the standard BCS relations; the
original relations are thus severely violated. In the opposite limit of an
opaque interface, the behavior of the system is in many respects close to the
BCS predictions. Over the entire range of \rho_{int}, the properties of the
bilayer are found numerically. Finally, it is shown that the results obtained
for the bilayer also apply to more complicated structures such as SNS and NSN
trilayers, SNINS and NSISN systems, and SN superlattices.
