This paper presents a systematic classification of symmetry protected topological (SPT) phases using group cohomology theory. SPT phases are gapped, short-range-entangled quantum phases protected by a symmetry group G. The paper shows that interacting bosonic SPT phases can be systematically described by group cohomology classes $ \mathcal{H}^{1+d}[G, U_T(1)] $, which label distinct d-dimensional SPT phases with on-site symmetry G. These phases are characterized by their boundary excitations, which are described by lattice non-linear σ-models with non-local Lagrangian terms. The paper also discusses the relationship between SPT phases and topological non-linear σ-models, showing that the boundary excitations of non-trivial SPT phases are gapless or degenerate. The paper provides explicit ground state wave functions and commuting projector Hamiltonians for SPT phases, and shows that the symmetry G must be realized as a non-on-site symmetry for the low energy boundary excitations. The paper also discusses the classification of SPT phases for various symmetry groups, including $ SO(3) $, $ SU(2) $, and $ U(1) $, and shows that the number of non-trivial SPT phases depends on the symmetry group and spatial dimension. The paper concludes that the results are much more general than previous examples, as they apply to any symmetry group. The paper also discusses the classification of short-range-entangled states with or without symmetry breaking, showing that these phases are labeled by three mathematical objects: $ (G_H, G_\Psi, \mathcal{H}^{1+d}[G_\Psi, U_T(1)]) $, where $ G_H $ is the symmetry group of the Hamiltonian and $ G_\Psi $ is the symmetry group of the ground states. The paper also discusses the relationship between group cohomology and Berry phase, and provides a detailed classification of SPT phases for various symmetry groups and spatial dimensions.This paper presents a systematic classification of symmetry protected topological (SPT) phases using group cohomology theory. SPT phases are gapped, short-range-entangled quantum phases protected by a symmetry group G. The paper shows that interacting bosonic SPT phases can be systematically described by group cohomology classes $ \mathcal{H}^{1+d}[G, U_T(1)] $, which label distinct d-dimensional SPT phases with on-site symmetry G. These phases are characterized by their boundary excitations, which are described by lattice non-linear σ-models with non-local Lagrangian terms. The paper also discusses the relationship between SPT phases and topological non-linear σ-models, showing that the boundary excitations of non-trivial SPT phases are gapless or degenerate. The paper provides explicit ground state wave functions and commuting projector Hamiltonians for SPT phases, and shows that the symmetry G must be realized as a non-on-site symmetry for the low energy boundary excitations. The paper also discusses the classification of SPT phases for various symmetry groups, including $ SO(3) $, $ SU(2) $, and $ U(1) $, and shows that the number of non-trivial SPT phases depends on the symmetry group and spatial dimension. The paper concludes that the results are much more general than previous examples, as they apply to any symmetry group. The paper also discusses the classification of short-range-entangled states with or without symmetry breaking, showing that these phases are labeled by three mathematical objects: $ (G_H, G_\Psi, \mathcal{H}^{1+d}[G_\Psi, U_T(1)]) $, where $ G_H $ is the symmetry group of the Hamiltonian and $ G_\Psi $ is the symmetry group of the ground states. The paper also discusses the relationship between group cohomology and Berry phase, and provides a detailed classification of SPT phases for various symmetry groups and spatial dimensions.