  We develop the theory of the spin and valley Hall effects in two-dimensional
systems caused by asymmetric -- skew -- scattering of the quasiparticles. The
collision integral is derived in the third order in the particle-particle
interaction with account for the spin-orbit coupling both for bosons and
fermions. It is shown that the scattering asymmetry appears only in the
processes where the interaction between the particles in the initial and
intermediate state is present. We show that for degenerate electrons or
nondegenerate particles the spin and valley currents induced by interparticle
collisions are suppressed with their steady-state values being proportional to
the squared temperature or density, respectively. Our results imply non-Fermi
liquid properties of electrons in the presence of electron-electron skew
scattering. Strong deviations from conventional picture of interparticle
scattering are also demonstrated for the skew scattering of two-dimensional
degenerate bosons, e.g. excitons or exciton polaritons: The spin or valley
current of degenerate bosons contains the enhancement factor exponentially
growing with increase in the particle density.
