The book "Computational Approach to Riemann Surfaces" provides a comprehensive overview of existing and developing computer-based approaches to Riemann surfaces. The contributions come from experts who provide publicly available numerical codes in this field. This illustrates which software tools are available and how they can be applied in practice. The work also includes examples of solutions to partial differential equations and in surface theory. It is aimed at both students pursuing a degree in the numerical methods of Riemann surfaces and professionals from geometry and mathematical physics who utilize the theory of Riemann surfaces in their research. This book is the first of its kind to reflect the advancements in this area over the past ten years and contains original results.