  Quantum transitions between the Mott insulator and metals by controlling
filling in two-dimensional square lattice are characterized by a large
dynamical exponent $z=4$ where the origin of unusual metallic properties near
the Mott insulator are ascribed to the proximity of the transition. The scaling
near the transition indicates the formation of flat dispersion area due to
singular momentum dependence of the single-particle renormalization. The flat
dispersion controls critical properties of the Mott transition. An instability
of the flat dispersion to the d-wave superconducting order is discussed. We
also discuss a case of the Mott transition for a model of Mn perovskite
compounds with orbital degeneracy where orbital correlation length shows
critical divergence toward the metal-insulator transition.
