Abstract:The usual notion of a Lie algebroid representation is known to be inadequate, in the sense that it does not include a natural adjoint representation. I will introduce VB--algebroids and show how these objects up to isomorphism are in one-to-one correspondence with certain Lie algebroid superconnections, up to a notion of equivalence; they can be seen as a generalization of the concept of Lie algebroid representations which do include an adjoint. Then I will describe the classification of VB-algebroids and a strange characteristic class that emerges from the analysis. This is joint work with Rajan A. Mehta.