This course is to provide prospective high school teachers in mathematics a rigorous treatment of the foundations of Euclidean geometry and an introduction to non-Euclidean geometry. Topics include: Euclid’s postulates, attempts to prove the parallel postulate, Hilbert’s axioms-a reformulation of the foundations of Euclidean geometry, geometry without the parallel postulate, the discovery of non-Euclidean geometry, consistency of hyperbolic geometry and models of hyperbolic geometry of Beltrami-Klein and Poincaré.