Composed by Edward Elgar, the Enigma Variations are a set of a theme and its fourteen variations written for orchestra in 1898–1899. It is Elgar's best-known large-scale composition, for both the music itself and the enigmas behind it. Elgar dedicated the piece to "my friends pictured within", each variation being an affectionate portrayal of one of his circle of close acquaintances.

The enigma is not the identity of the persons portrayed, as those are known, but rather a hidden theme that is, in Elgar's words, "not played". This hidden theme has been the subject of much speculation, and various musicians have proposed theories for what melody it could be, although Elgar did not say that it was a melody. The enigma could be something else, such as a symbol or a literary theme. Elgar accepted none of the solutions that were put forward in his lifetime, and, pleased with his little joke, took the secret with him to the grave.

See: musical cryptogram.

The Wikipedia page - which lists the variations - is [1]. There is much coverage elsewhere - the Elgar website [2] and BBC Radio 3 [3]. There are also various claims of solving the enigma.