  We present a comprehensive quasiclassical approach for studying transport
properties of superconducting diffusive hybrid structures in the presence of
extrinsic spin-orbit coupling. We derive a generalized Usadel equation and
boundary conditions that in the normal state reduce to the drift-diffusion
theory governing the spin-Hall effect in inversion symmetric materials. These
equations predict the non-dissipative spin-galvanic effect, that is the
generation of supercurrents by a spin-splitting field, and its inverse -- the
creation of magnetic moment by a supercurrent. These effects can be seen as
counterparts of the spin-Hall, anomalous Hall and their inverse effects in the
superconducting state. Our theory opens numerous possibilities for using
superconducting structures in magnetoelectronics.
