I will talk about the classification of tensor ideals in the Deligne categories attached to the symmetric, orthogonal and general linear group and give some applications.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/8057