  We review the half-Skyrmion theory for copper-oxide high-temperature
superconductivity. In the theory, doped holes create a half-Skyrmion spin
texture which is characterized by a topological charge. The formation of the
half-Skyrmion is described in the single hole doped system, and then the
half-Skyrmion excitation spectrum is compared with the angle-resolved
photoemission spectroscopy results in the undoped system. Multi-half-Skyrmion
configurations are studied by numerical simulations. We show that
half-Skyrmions carry non-vanishing topological charge density below a critical
hole doping concentration \sim 30% even in the absence of antiferromagnetic
long-range order. The magnetic structure factor exhibits incommensurate peaks
in stripe ordered configurations of half-Skyrmions and anti-half-Skyrmions. The
interaction mediated by half-Skyrmions leads to d_{x^2-y^2}-wave
superconductivity. We also describe pseudogap behavior arising from the
excitation spectrum of a composite particle of a half-Skyrmion and doped hole.
