1 Introduction: Surfaces with prescribed mean curvature -- 2 From minimal surfaces and CMC surfaces to harmonic maps -- 2.1 Minimal surfaces -- 2.2 Constant mean curvature surfaces -- 3 Variational point of view and Noetherโ{128}{153}s theorem -- 4 Working with the Hopf differential -- 4.1 Appendix -- 5 The Gauss-Codazzi condition -- 5.1 Appendix -- 6 Elementary twistor theory for harmonic maps -- 6.1 Appendix -- 7 Harmonic maps as an integrable system -- 7.1 Maps into spheres -- 7.2 Generalizations -- 7.3 A new setting: loop groups -- 7.4 Examples -- 8 Construction of finite type solutions -- 8.1 Preliminary: the Iwasawa decomposition (for). -- 8.2 Application to loop Lie algebras -- 8.3 The algorithm -- 8.4 Some further properties of finite type solutions -- 9 Constant mean curvature tori are of finite type -- 9.1 The result -- 9.2 Appendix -- 10 Wente tori -- 10.1 CMC surfaces with planar curvature lines -- 10.2 A system of commuting ordinary equations -- 10.3 Recovering a finite type solution -- 10.4 Spectral curves -- 11 Weierstrass type representations -- 11.1 Loop groups decompositions -- 11.2 Solutions in terms of holomorphic data -- 11.3 Meromorphic potentials -- 11.4 Generalizations