The gauge field theory of defects as a newly-developed and emerging branch in modern solid mechanics and material science is reviewed in detail. Noether's theorem and its inverse theorem, which play a significant role in the construction of the defect gauge theory is first introduced. The original works and contributions due to Golebicwska-Lasota, Edelen, Kadic and Edelen, etc, are systematically brought in on the basis of Yang-Mills universal gauge theory construction, including the minimal replacement pri...