Lévy processes and infinitely divisible distributions are increasingly defined in terms of their Lévy measure. In order to describe the dependence structure of a multivariate Lévy measure, Tankov (2003) introduced positive Lévy copulas. Together with the marginal Lévy measures they completely describe multivariate Lévy measures on $\mathbb{R}^m_+$. In this paper, we show that any such Lévy copula defines itself a Lévy measure with 1-stable margins, in a canonical way. A limit theorem is obtained, characterising convergence of Lévy measures with the aid of Lévy copulas. Homogeneous Lévy copulas are considered in detail. They correspond to Lévy processes which have a time-constant Lévy copula. Furthermore, we show how the Lévy copula concept can be used to construct multivariate distributions in the Bondesson class with prescribed margins in the Bondesson class. The construction depends on a mapping $\Upsilon$, recently introduced by Barndorff-Nielsen and Thorbjørnsen (2004a,b) and Barndorff-Nielsen, Maejima and Sato (2004). Similar results are obtained for self-decomposable distributions and for distributions in the Thorin class.