Harmonic Maps Between Surfaces (with a Special Chapter on Conformal Mappings) Geometric considerations Conformal mappings Existence theorems for harmonic maps between surfaces Uniqueness theorems A priori C1  estimates A priori estimates from below for the functional determinant of harmonic diffeomorphisms The existence of harmonic diffeomorphisms which solve a Dirichlet problem C1  a priori estimates for arbitrary domains. Non variational existence proofs Harmonic coordinates. C2  a priori estimates for harmonic maps The existence of harmonic diffeomorphisms between surfaces Applications of harmonic maps between surfaces.