Explore the computational theory of Riemann-Hilbert problems in this comprehensive lecture by Thomas Trogdon at the International Centre for Theoretical Sciences. Delve into key concepts including simple Riemann-Hilbert problems, function definitions, properties of Psi, Cauchy integrals, and analytical functions. Examine important facts and classifications related to the topic. Part of a broader program on integrable systems in mathematics, condensed matter, and statistical physics, this lecture provides a solid foundation for understanding the computational aspects of Riemann-Hilbert problems and their applications in various fields of mathematics and physics.