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I review and discuss the construction of supersymmetry multiplets andmanifestly supersymmetric Batalin-Vilkovisky actions using pure spinors, withemphasis on models with maximal supersymmetry. The special cases of D=3, N=8(Bagger-Lambert-Gustavsson) and N=6 (Aharony-Bergman-Jafferis-Maldacena)conformal models are treated in detail. Most of the material is covered by thepapers arXiv:0808.3242 and arXiv:0809.0318. This is the written version of atalk given at 4th Baltic-Nordic workshop "Algebra, Geometry and MathematicalPhysics", Tartu, Estonia, October 9-11, 2008, to appear in the Proceedings ofthe Estonian Academy of Sciences, vol 4, 2010.

Pure spinor superfields, with application to D=3 conformal models