This advanced reading course explores José Fernando Escobar's groundbreaking work on the boundary Yamabe problem, a fundamental question in differential geometry concerning the existence of constant mean curvature metrics on Riemannian manifolds with boundary. We will begin with necessary background in Riemannian geometry, conformal transformations, and the variational structure underlying the Yamabe functional. The course will cover Escobar's existence and uniqueness results, particularly his resolution of the problem under certain geometric conditions and his construction of the boundary Yamabe invariant. Key mathematical tools include blow up analysis, concentration-compactness principles, and the method of moving planes. Students should seek the course instructor’s approval to take this course.