  We define the ordinary Minkowski space inside the conformal space according
to Penrose and Manin as homogeneous spaces for the Poincar\'e and conformal
group respectively. We realize the supersymmetric (SUSY) generalizations of
such homogeneous spaces over the complex and the real fields. We finally
investigate chiral (antichiral) superfields, which are superfields on the super
Grassmannian, Gr(2|1, 4|1), respectively on Gr(2|0, 4|1). They ultimately give
the twistor coordinates necessary to describe the conformal superspace as the
flag Fl(2|0, 2|1; 4|1) and the Minkowski superspace as its big cell.
