(Top) Schematic illustration of lattice bosons near unit filling. At n=1 in the Mott insulating phase (MI) when (J/U)<(J/U)c the effective wave function of each particle spreads only a finite distance ξ(J). For n>1 (n<1), the extra particles (holes) travel freely within the essentially inert background, and the state is a superfluid (SF) for arbitrarily small |n-1|. (Bottom) Phase diagram, showing Mott insulating (including those with higher integer filling) and SF phases. The transitions through the tips of the Mott lobes at (J/U)c (and following the dashed lines at constant filling beyond them) have a special particle-hole symmetry.