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Twentieth century mathematics can be characterized by the study of functions. The most interesting functions are those that preserve the structure at hand. In many branches of mathematics transformations are looked at and classified. In geometry, transformations of a plane are very important. A transformation that distance is called an isometry. The classification of isometries is known in Euclidean geometry, and one will learn about each of them in any geometry course. Classification of isometries for the hyperbolic plane are not as well known, but they have been classified. In this paper we will discover hyperbolic geometry, investigate some of the models used to view hyperbolic geometry and discuss some of the isometries of the hyperbolic plane.