Niels Henrik Abel (1802–29) was one of the most prominent mathematicians in the first half of the nineteenth century. His pioneering work in diverse areas such as algebra, analysis, geometry and mechanics has made the adjective 'abelian' a commonplace in mathematical writing. These collected works, first published in two volumes in 1881 after careful preparation by the mathematicians Ludwig Sylow (1832–1918) and Sophus Lie (1842–99), contain some of the pillars of mathematical history. Volume 1 includes perhaps the most famous of Abel's results, namely his proof of the 'impossibility theorem', as well as his 'Paris memoir', which contains his many fundamental results on transcendental functions. Volume 2 contains additional articles on elliptic functions and infinite series. It also includes extracts from Abel's letters, as well as detailed notes and commentary by Sylow and Lie on Abel's pioneering work.