  In this talk, we review how the superspace formulation of maximally
supersymmetric field theories (including supergravity) naturally leads to
introduction of pure spinors and pure spinor superfields, and why the formalism
provides off-shell formulations. This approach to pure spinor superfields thus
stresses field-theoretic aspects rather than the first-quantised ones normally
used e.g. in superstring theory. We discuss how the BRST operator arises and
the principles behind constructions of actions, as well as the general
Batalin-Vilkovisky framework. D=11 supergravity and its recently constructed
supersymmetric action are taken as an example throughout the talk. This is the
written version of a lecture given at the 6th Mathematical Physics Meeting,
Belgrade, September 2010.
