Of the 26 sporadic finite simple groups, 5 were discovered by E. Mathieu in 1861 and 1873 [1], [2]. These Mathieu groups are the focus of this thesis, where we will prove their simplicity using elementary methods. E. Witt [5] realized a connection between the Mathieu groups and certain combinatorial structures known as Steiner systems. We will follow his construction to define the Mathieu groups as the automorphism groups of certain Steiner systems. Much of the work of the thesis lies in the construction of these Steiner systems, which we achieve by using both methods from finite geometry and the theory of Golay codes.