The Farey Graph is a beautiful mathematical object. As a metric space, it is hyperbolic. It is built from the rational numbers and is acted upon by the group \(SL_2 (\mathbb{Z})\). In this talk I would like to show how the study of the Farey Graph combines topology, group theory and number theory. There will be no ground breaking results, but the Farey Graph is an interesting object because of its simplicity and the many fascinating connections it provides. We will study hyperbolicity, relatively hyperbolic groups, the boundary at infinity, the Euclidean algorithm, a fool's way to add rational numbers, \(2 \times 2\) matrices, geodesics and the real numbers.