Monopoles and Quark Confinement

I review the studies of quark confinement based on the dual Meissner effect due to monopole condensation after abelian projection in QCD. The first part is about Monte Carlo simulations of abelian projection and the role of monopoles in lattice QCD. Abelian projection in the maximally abelian gauge is found very interesting. The monopole part alone is responsible for confinement. A block spin transformation on the dual lattice and energy-entropy balance of the monopole loops suggest that lattice SU(2) QCD is always (for all β) in the monopole condensed phase and so in the confinement phase in the infinite volume limit. Abelian Polyakov loops in various gauges suggest gauge independence of the picture of the monopole condensation. The effective field theory composed of a dual photon and a monopole field based on the monopole condensation after abelian projection is shortly reviewed in the next part.