This book provides an accessible and self-contained introduction to harmonic map theory and its analytical aspects, covering recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. The reader is then presented with a guided tour of the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions.

A self-contained presentation of "exotic" functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a "Coulomb moving frame" is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to theory of surfaces.

1. Geometric and analytic setting
2. Harmonic maps with symmetry
3. Compensations and exotic function spaces
4. Harmonic maps without symmetry
5. Surfaces with mean curvature in L