Elliptic problems arise in the study of significant questions concerning vortices in various selfdual field theories, e.g., Chern - Simons and Electroweak theories. Despite progress in these directions, many open questions still remain and are examined in this work in connection with Liouville-type equations and systems.Key topics unfold systematically beginning with the foundations of gauge theory and examples of selfdual vortices; chapters thereafter treat elliptic problems involving selfdual vortices, Ginzburg - Landau and Chern - Simons models, concentration-compactness principles, and Maxwell - Chern - Simons vortices.To date there is no comprehensive examination of the subject. The primary orientation of the text is partial differential equations, but the reader should have knowledge of some basic tools in field theory. For graduate students and researchers in PDEs and mathematical physics.