|J. Grabowski, K. Grabowska and I have placed a preprint on the arXiv called ‘Introduction to Graded Bundles‘ , which is based on a talk given by Prof Grabowski at the First International Conference of Differential Geometry, Fez (Morocco), April 11-15, 2016.|
The preprint outlines much of our recent work on graded bundles (a nice kind of graded manifold) and their linearisation (as a functor to k-fold vector bundles), as well as the notions of weighted Lie groupoids and algebroids, including the Lie theory.
One key observation that must be made is that there are many examples of graded bundles that appear in the existing literature, it is just that they are not recognised as such and their graded structure is not really exploited. The canonical example here are the higher order tangent bundles which are well studied from the perspective of higher order mechanics.
Anyway, if anyone want to get a quick overview of some of the ideas behind my work, then I direct them to this preprint. If you are interested in the applications to mechanics, then I suggest  as well as references therein.
 Introduction to graded bundles, Andrew J. Bruce, K. Grabowska, J. Grabowski, arXiv:1605.03296 [math.DG]
 New developments in geometric mechanics, A. J. Bruce, K. Grabowska, J. Grabowski, P. Urbanski, arXiv:1510.00296 [math-ph].