Sobolev Maps to the Circle : From the Perspective of Analysis Geometry and Topology Lifting in $W^{1 p}$ The Geometry of $J(u)$ and $\\Sigma(u)$ in 2D; Point Singularities and Minimal Connections The Geometry of $J(u)$ and $\\Sigma(u)$ in 3D (and higher); Line Singularities and Minimal Surfaces A Digression: Sphere Valued Maps Lifting in Fractional Sobolev Spaces and in $VMO$ Uniqueness of Lifting and Beyond Factorization Applications of the Factorization Estimates of Phases: Positive and Negative Results Density Traces Degree Dirichlet Problems Gaps Infinite Energies Domains with Topology Appendices. The theory of real valued Sobolev functions is a classical part of analysis and has a wide range of applications in pure and applied mathematics. By contrast the study of manifold valued Sobolev maps is relatively new. The incentive to explore these spaces arose in the last forty years from geometry and physics. This monograph is the first to provide a unified comprehensive treatment of Sobolev maps to the circle presenting numerous results obtained by the authors and others. Many surprising connections to other areas of mathematics are explored including the Monge Kantorovich theory in optimal transport items in geometric measure theory Fourier series and non local functionals occurring for example as denoising filters in image processing. Numerous digressions provide a glimpse of the theory of sphere valued Sobolev maps. Each chapter focuses on a single topic and starts with a detailed overview followed by the most significant results and rather complete proofs. The “Complements and Open Problems” sections provide short introductions to various subsequent developments or related topics and suggest new directions of research. Historical perspectives and a comprehensive list of references close out each chapter. Topics covered include lifting point and line singularities minimal connections and minimal surfaces uniqueness spaces factorization density Dirichlet problems trace theory and gap phenomena. Sobolev Maps to the Circle will appeal to mathematicians working in various areas such as nonlinear analysis PDEs geometric analysis minimal surfaces optimal transport and topology. It will also be of interest to physicists working on liquid crystals and the Ginzburg Landau theory of superconductors.