  The paper investigates contributions of $Z$, $W$ and $\gamma$ intermediate
states to the parity-violating Moller scattering asymmetry up to two-loop
level. Using the Yennie--Frautschi--Suura factorization form for amplitudes, we
demonstrate that QED corrections, with an exception of vacuum-polarization
effects, cancel at the asymmetry level. We obtain chiral amplitudes at Born,
one-loop and partially at two-loop level: boxes with lepton self-energies,
ladder boxes and decorated boxes. Our calculations are relevant for the
ultra-precise 11 GeV MOLLER experiment planned at Jefferson Laboratory and
future ILC experiments. The numerical comparision of the two-loop contributions
with the experimental accuracy of MOLLER is provided.
