It is difficult for a high school student to fully appreciate the study of Euclidean geometry without some notion of its relation to other geometries and to the space in which we live. Too often, he regards the geometry of Euclid with near reverence, as one of the few things that is completely definite. For this reason, an attempt will be made to point out the development and basic char- . acteristics of the Non-Euclidean geometries of Lobachevsky and Riemann, which Klein designated as Hyperbolic and Elliptic geometry, respectively. Due to the limitations of the author and to the fact that this may, at some time, be presented at the high school level, no rigorous development will be attempted. Instead, a logical sequence of the elementary principles underlying Non-Euclidean geometry will be presented in the hope that a reader may gain some concept, at least, of the Non-Euclidean geometries.