Riemann-Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann-Hilbert problem. This book provides introductions to both computational complex analysis, as well as to the applied theory of Riemann-Hilbert problems from an analytical and numerical perspective. Following a full-discussion of applications to integrable systems, differential equations and special function theory, the authors include six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann-Hilbert method, each of mathematical or physical significance, or both. As the most comprehensive book to date on the applied and computational theory of Riemann-Hilbert problems, this book is ideal for graduate students and researchers interested in a computational or analytical introduction to the Riemann-Hilbert method.