2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/138780This text is meant to be a brief overview of the topics announced in the title and is based on my talk in Vienna (August/September 2007). It does not contain new results (except probably for a remark concerning Q-manifold homology, which I wish to elaborate elsewhere). "Mackenzie theory" stands for the rich circle of notions that have been put forward by Kirill Mackenzie (solo or in collaboration): double structures such as double Lie groupoids and double Lie algebroids, Lie bialgebroids and their doubles, nontrivial dualities for double and multiple vector bundles, etc. "Q-manifolds" are (super)manifolds with a homological vector field, i.e., a self-commuting odd vector field. They may have an extra Z-grading (called weight) not necessarily linked with the Z_2-grading (parity). I discuss double Lie algebroids (discovered by Mackenzie) and explain how this quite complicated fundamental notion is equivalent to a very simple one if the language of Q-manifolds is used. In particular, it shows how the two seemingly different notions of a "Drinfeld double" of a Lie bialgebroid due to Mackenzie and Roytenberg respectively, turn out to be the same thing if properly understood.LaTeX, 17 pages; based on a talk at ESI, August/September 2007Differential GeometryMathematical PhysicsSymplectic GeometryPrimary 53D17. Secondary 17B62, 17B66, 18D05, 58A50, 58C50, 58H05text