Hilbert's 12th problem (Kronecker's Jugendtraum) is one of the major open problems
in number theory. In its various forms it asks for an explicit construction of abelian extensions of a given number field F and related arithmetical ingredients such as units in these extensions, or an explicit description of their Galois group. The only fully developed cases are when F is the field of rational numbers (cyclotomy) or a quadratic imaginary field (complex multiplication). After a brief survey of these classical cases we shall describe some amazing developments, still largely conjectural, by Darmon and his collaborators (Dasgupta, Vonk) when F is a real quadratic field.