1. Historical. - 2. New metric. - 3. The fundamental A- B- and C-functions. - 4. Method of areal proximity. - 5. Summary. - I Mappings Into Closed Riemann Surfaces. - 1. Mappings of Arbitrary Riemann Surfaces. - 2. Meromorphic Functions on Arbitrary Riemann Surfaces. - 3. Surfaces RS and Conformal Metrics. - II Mappings Into Open Riemann Surfaces. - 1. Principal Functions. - 2. Proximity Functions on Arbitrary Riemann Surfaces. - 3. Analytic Mappings. - III Functions of Bounded Characteristic. - 1. Decomposition. - 2. The Class OMB. - IV Functions on Parabolic Riemann Surfaces. - 1. The Evans-Selberg Potential. - 2. Meromorphic Functions in a Boundary Neighborhood. - V Picard Sets. - 1. Infinite Picard Sets. - 2. Finite Picard Sets. - VI Riemannian Images. - 1. Mean Sheet Numbers. - 2. Euler Characteristic. - 3. Islands and Peninsulas. - 4. Meromorphic Functions. - 5. Mappings of Arbitrary Riemann Surfaces. - Appendix I. Basic Properties of Riemann Surfaces. - Appendix II. Gaussian Mapping of Arbitrary Minimal Surfaces. - 1. Triple connectivity. - 2. Arbitrary connectivity. - 3. Arbitrary genus. - 4. Arbitrary genus and connectivity. - 5. Gaussian mapping. - 6. Picard directions. - 7. Islands and peninsulas. - 8. Regular exhaustions. - 9. Open questions. - Author Index. Language: English