Higher order algebroids are generalizations of higher order tangent
bundles and Lie algebroids at the same time. They appear naturally in
the context of geometric mechanics when higher order derivatives and
symmetry are in the game. In the approach of M. Jóźwikowski and
M. Rotkiewicz they are introduced by means of a vector bundle comorphism
of a special kind. Natural examples come from reductions of higher
order tangent bundles of groupoids. I will explain the algebraic
structure staying behind higher order Lie algebroids, at least in
order two. It turned out that they lead to representations up to
homotopy of Lie algebroids, a fundamental notion in the theory of
algebroids discovered by C. A. Abad and M. Crainic.