Euler made many contributions to what is now called combinatorics. Some of these arose from recreational mathematics, such as magic squares and knight's tours on a chessboard; others from his study of lotteries; and, perhaps his most important work, from the study of partitions of an integer. In what follows I shall attempt to show the breadth of his work. Euler's papers will be referred to by their Eneström numbers, such as E338; this cataloguing was carried out in the early twentieth century by the Swedish Mathematician Gustav Eneström, who was to Euler what Köchel has been to Mozart. Euler's papers can all be studied in detail on the Euler Archive, details of which are given at the end of this article.