  In oder to investigate quark confinement, we give a new reformulation of the
$SU(N)$ Yang-Mills theory on a lattice and present the results of the numerical
simulations of the $SU(3)$ Yang-Mills theory on a lattice. The numerical
simulations include the derivation of the linear potential for static
interquark potential, i.e., non-vanishing string tension, in which the
"Abelian" dominance and magnetic monopole dominance are established,
confirmation of the dual Meissner effect by measuring the chromoelectric flux
tube between quark-antiquark pair, the induced magnetic-monopole current, and
the type of dual superconductivity, etc.
