In modern theoretical physics, gauge field theories are of great importance since they keep internal symmetries and account for phenomena such as spontaneous symmetry breaking, the quantum Hall effect, charge fractionalization, superconductivity and supergravity. This monograph discusses specific examples of selfdual gauge field structures, including the ChernSimons model, the abelianHiggs model, and YangMills gauge field theory.The author builds a foundation for gauge theory and selfdual vortices by introducing the basic mathematical language of gauge theory and formulating examples of ChernSimonsHiggs theories (in both abelian and non-abelian settings). Thereafter, the electroweak theory and self-gravitating electroweak strings are examined. The final chapters treat elliptic problems involving ChernSimmons models, concentration-compactness principles, and MaxwellChernSimons vortices.Many open questions still remain in the field and are examined in this work in connection with Liouville-type equations and systems. The goal of this text is to form an understanding of selfdual solutions arising in a variety of physical contexts and thus is ideal for graduate students and researchers interested in partial differential equations and mathematical physics.