Explore hyperbolic geometry, the modular group, and Diophantine approximation in this comprehensive lecture from the Geometry, Groups and Dynamics program at the International Centre for Theoretical Sciences. Delve into key concepts such as the hyperbolic plane, its boundary, subgroups of SL(2,R), and the unit tangent bundle. Examine the geodesic flow, fundamental domains, and the Dirichlet fundamental domain. Learn about important propositions and proofs related to these topics. This 1-hour 14-minute lecture serves as an in-depth introduction to these advanced mathematical concepts, suitable for graduate students and researchers in geometry, dynamical systems, and group theory.