This book expands the author's Quantum Space Theory: a Quantum Field Theory of Particles containing internal spaces. Fermion, scalar boson vector boson, and graviton space particles are formulated. Spinor indices and square spinor arrays play a crucial role in the theory. They lead to the dimension arrays of the ten spaces of Octonion Cosmology. The unitary symmetry group on spinor indices has both global and local formulations. We show that they lead to a map from square spinor arrays to arrays of dimensions for octonion spaces. The Octonion Spaces Spectrum is derived from a 20 space-time dimension point (thus giving upper limit to the octonion spaces spectrum). The spectrum is generated by repeated fermion-antifermion annihilations. The Perturbation Theory of Quantum Space Theory is developed and applied to the creation of an octonion space from fermion-antifermion annihilation. Analysis of spinor arrays leads to blocks of 16 fundamental fermions; and blocks of internal symmetries with irreducible representations totaling 16 dimensions. The blocking of 64 dimensions into fermion layers is also derived. Quantum Space Theory leads (via Octonion Cosmology) to the author's NEWQUeST, NEWUTMOST, and NEWUST. It provides a complete derivation of Octonion Cosmology yielding the known features of elementary particles. The author's Theory of Everything is thereby based on the Quantum Field Theory of Spaces. It has a clear, direct line of development based on the author's papers in the 1970's, and books in the Twenty-First Century. The book suggests experimental tests of Octonion Cosmology. The book discusses the possible relation of the Octonion Spaces spectrum to the various SuperString theories. Lastly, the book indicates the possible application of Quantum Space Theory to Fermionic Condensed Matter.