A double vector bundle is a vector bundle object in the category of vector bundles. The iterated tangent bundle and, more generally, the tangent bundle of a vector bundle, have played a role in some approaches to connection theory for many years, and various combinations of tangent and cotangent structures are basic to global classical mechanics.

Somewhat surprisingly, dualizing one structure in a double vector bundle leads to a second double vector bundle. Further, the two duals obtained in this way are themselves dual. This leads to a `three--step' duality which appears to be an entirely new
phenomena. The talk will describe this, and demonstrate that it arises naturally from considering triple structures.