We provide an account of Milnor's construction of an exotic 7-sphere and the subsequent rapid development of differential topology used to produce and classify exotic manifolds. We begin by giving some necessary background, assuming only previous knowledge of real analysis and linear algebra. Smooth manifolds, vector bundles, and fibre bundles are introduced, along with some operations on manifolds not usually seen in a first course, before giving a review of all necessary algebraic topology. We pay particular attention to the homology of smooth manifolds, as this will form the basis for the constructions of the following section. We then introduce characteristic classes, which are one of the main ingredients in constructing smooth manifold invariants. Using this setup, we develop Milnor's original smooth invariant and a generalisation of it to a wider class of manifolds. We give a brief introduction to Morse theory, which we use to characterise topological spheres. Having set up the necessary background, we construct a number of examples of exotic spheres. We first present Milnor's original example, and then develop a more general tool, plumbing disk bundles, to give a much larger class of examples. Finally, we turn to the classiffication of smooth structures on spheres of dimension greater than four, developing the necessary background to state Milnor and Kervaire's classiffication results on homotopy groups of spheres, before indicating a number of future directions of study to the reader, as this thesis is ultimately intended to be an introduction to a vast field.