  We interpret superfields in a functorial formalism that explains the
properties that are assumed for them in the physical applications. The starting
point of this research was the need to understand in a sound mathematical
framework some algebraic constraints imposed on them, but it lead us to revise
the very definition of superfield. The constraints that we investigate in the
present work give rise to superschemes that, generically, are not regular, that
is, they do not define a standard supermanifold.
