Explore the fourth lecture in a series on Rational Points on Modular Curves, delivered by Jan Vonk at the International Centre for Theoretical Sciences. Delve into advanced topics in arithmetic geometry, focusing on the study of rational points on modular curves. Gain insights into the geometry of modular curves, their Q-rational points, and both classical and non-abelian Chabauty methods. Discover the theoretical and computational aspects of determining K-rational points on modular curves XH(K) for various fields K and subgroups H of GL2(Z/NZ). Engage with a balanced presentation of advanced concepts and practical examples in this 1 hour and 34 minute lecture, part of a comprehensive program designed to foster collaboration among experts and encourage participation from young researchers in the field of arithmetic geometry.