Foundations. Linear functions: 1.1 Analytic functions and conformal mapping; 1.2 Integral linear functions; 1.3 The function $w=1\slash z$; 1.3a Appendix to 1.3: Stereographic projection; 1.4 Linear functions; 1.5 Linear functions (continued); 1.6 Groups of linear functions Rational Functions: 2.7 $w=z^n$; 2.8 Rational functions General considerations: 3.9 The relation between the conformal mapping of the boundary and that of the interior of a region; 3.10 Schwarz' principle of reflection Further study of mappings represented by given formulas: 4.11 Further study of the geometry of $w=z^2$; 4.12 $w=z+1\slash z$; 4.13 The exponential function and the trigonometric functions; 4.14 The elliptic integral of the first kind Mappings of given regions: 5.15 The mapping of a given region onto the interior of a circle (illustrative examples); 5.16 Vitali's theorem on double series; 5.17 A limit theorem for simple mappings; 5.18 Proof of Riemann's mapping theorem; 5.19 On the actual construction of the conformal mapping of a given region onto a circular disc; 5.20 Potential-theoretic considerations; 5.21 The correspondence between the boundaries under conformal mapping; 5.22 Distortion theorems for simple mappings of the disc $\vert z\vert 1$; 5.24 On the conformal mapping of non-simple, simply-connected regions onto a circular disc; 5.24a Remark on the mapping of non-simple, multiply-connected regions onto simple regions; 5.25 The problems of uniformization; 5.26 The mapping of multiply-connected plane regions onto canonical regions Bibliography Index