The Bose-Hubbard Hamiltonian H = -J Σ(a†ᵢaⱼ + h.c.) + U/2 Σnᵢ(nᵢ-1) - μΣnᵢ describes bosons on a lattice. At J/U ≫ 1: superfluid (delocalized). At J/U ≪ 1: Mott insulator (integer filling, gap). The phase boundary forms Mott lobes in (J/U, μ/U) space.