  A VB-algebroid is essentially defined as a Lie algebroid object in the
category of vector bundles. There is a one-to-one correspondence between
VB-algebroids and certain flat Lie algebroid superconnections, up to a natural
notion of equivalence. In this setting, we are able to construct characteristic
classes, which in special cases reproduce characteristic classes constructed by
Crainic and Fernandes. We give a complete classification of regular
VB-algebroids, and in the process we obtain another characteristic class of Lie
algebroids that does not appear in the ordinary representation theory of Lie
algebroids.
