We present three characterizations of n-dimensional Archimedean copulas: algebraic, differential and diagonal. The first is due to Jouini and Clemen. We formulate it in a more general form, in terms of an n-variable operation derived from a binary operation. The second characterization is in terms of first order partial derivatives of the copula. The last characterization uses diagonal generators, which are "regular'' diagonal sections of copulas, enabling one to recover the copulas by means of an asymptotic representation.