This paper explores the classification and construction of symmetry-protected topological (SPT) phases in interacting bosonic systems. SPT phases are gapped, short-range-entangled quantum phases that remain stable even when the symmetry is broken. The authors show that these phases can be systematically described using group cohomology theory, specifically by elements in $\mathcal{H}^{1+d}[G, U_T(1)]$, where $G$ is the symmetry group and $U_T(1)$ is the time-reversal symmetry group. They demonstrate that distinct $d$-dimensional SPT phases with on-site symmetry $G$ can be labeled by elements in this cohomology group. The paper also introduces a new type of topological term that generalizes the $\theta$-term in continuous non-linear $\sigma$-models to lattice non-linear $\sigma$-models, which is crucial for describing the boundary excitations of non-trivial SPT phases. The boundary states of these phases are gapless or degenerate, depending on the symmetry realization. The authors apply their theory to construct interacting bosonic topological insulators and superconductors, providing explicit ground state wave functions and commuting projector Hamiltonians. They also discuss the classification of SPT phases with translation symmetry and the relation between cocycles and Berry phases. The results are applicable to any symmetry group and provide a comprehensive framework for understanding SPT phases in interacting bosonic systems.This paper explores the classification and construction of symmetry-protected topological (SPT) phases in interacting bosonic systems. SPT phases are gapped, short-range-entangled quantum phases that remain stable even when the symmetry is broken. The authors show that these phases can be systematically described using group cohomology theory, specifically by elements in $\mathcal{H}^{1+d}[G, U_T(1)]$, where $G$ is the symmetry group and $U_T(1)$ is the time-reversal symmetry group. They demonstrate that distinct $d$-dimensional SPT phases with on-site symmetry $G$ can be labeled by elements in this cohomology group. The paper also introduces a new type of topological term that generalizes the $\theta$-term in continuous non-linear $\sigma$-models to lattice non-linear $\sigma$-models, which is crucial for describing the boundary excitations of non-trivial SPT phases. The boundary states of these phases are gapless or degenerate, depending on the symmetry realization. The authors apply their theory to construct interacting bosonic topological insulators and superconductors, providing explicit ground state wave functions and commuting projector Hamiltonians. They also discuss the classification of SPT phases with translation symmetry and the relation between cocycles and Berry phases. The results are applicable to any symmetry group and provide a comprehensive framework for understanding SPT phases in interacting bosonic systems.