  We give a short review of recently obtained results on a new lattice
formulation of the non-linear change of variables which was once called the
Cho--Faddeev--Niemi decomposition in SU(2) Yang-Mills theory.
  Based on this formulation, we proposed a new gauge-invariant definition of
the magnetic monopole current which guarantees the magnetic charge
quantization. We also demonstrated the magnetic monopole dominance in the
string tension in SU(2) Yang-Mills theory on a lattice. Our formulation enables
one to reproduce in the gauge-invariant way remarkable results obtained so far
only in the Maximally Abelian gauge.
