This is a draft version of a working paper on enveloping Leibniz dialgebras of Leibniz algebras. We show that these objects share many properties with universal enveloping algebras of Lie algebras, in particular a generalised bialgebra structure, PBW and Milnor-Moore type structure theorems, and a homological property that should be satisfied by formal objects integrating Leibniz algebras ("coquecigrues").