Electrons are particles with spin 1/2, with their time-reversal symmetry T satisfies T²=-1. But spinless particles and spin-1 particles, such as photons and phonons, have their time-reversal symmetry satisfying T²=1. The difference leads to distinct topological classifications of band theory with respect to dimensions. In this talk, I will first review basics of K-theoretical classification of topological band structures, and then focus on the case of band theory with T²=1 to demonstrate the abstract K-theoretical paradigm by discussing the corresponding topological classifications, characterizations and models. In particular, if inversion symmetry P is preserved, the band theory becomes a real band theory, namely, that at each point in momentum space the Hilbert space is essentially real by spacetime inversion symmetry PT. Consequently, the little group at each point in the Brillouin zone is represented by real matrices, where the irreducible blocks are classified into three classes: real, complex and symplectic, according to Freeman Dyson’s classic threefold-way paper in 1962. Surprisingly, new complex structure can arise in the real band theory, and counterpart of magnetic monopoles can be realized, but with some different properties.