I will briefly review Kontsevich's formality theorem and its generalization to graded manifolds. Then I will explain a potential application of the latter that was proposed by Cattaneo. This leads one to consider a certain super-geometric extension problem for Poisson structures. I will finally describe some simple examples where this extension problem can be solved.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/4234
[3] http://www.mpim-bonn.mpg.de/node/3946