This is a very thorough and unifying treatment of most methods for computing conformal maps from \({\mathbb{D}}\) onto a Jordan region. If \(\theta\) denotes the boundary correspondence function and \(\tau (t)=\theta (t)-t\), starting from a rather general auxiliary function H the author derives an operator equation \(\psi \tau =0\) where \(\psi\) depends on H. This can be solved by direct iteration or by the Newton method. Special choices of H give the classical methods associated with the names Theodorsen, Timman, Friberg and others. In the iterations to all these methods the conjugation operator K plays a key role which can be implemented by two fast Fourier transforms. The solution of \(\psi \tau =0\) by the Newton method leads in each step to a Riemann-Hilbert problem for the unit disc which can be solved explicitly by two operations K. Again specializing, the recent methods of Wegmann and Vertgeim-Hübner are obtained which converge locally quadratically. Further related methods are mentioned, and there is an extensive list of references.