This entry is about the concept in supergeometry. For the concept in gravity/cosmology see at Wheeler superspace.
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Physicists often refer to spaces in supergeometry, such as supermanifolds or super schemes, as superspaces, regardless the formalism. Hence a superspace can be an affine superspace (the affine counterpart of the supervector space over real or complex numbers), superscheme, supermanifold (of either Leites or de Witt kind) etc.
Mostly however “superspace” is used for superspacetimes (e.g. super Minkowski spacetimes), presented in either formalism.
The concept of superspace in physics (together with that of superfields) is due to
Abdus Salam J.A. Strathdee, Supergauge Transformations, Nucl.Phys. B76 (1974) 477-482 (spire)
Abdus Salam J.A. Strathdee, Physical Review D11, 1521-1535 (1975)
(which considered superspace if dimension $d = 4$ with number of supersymmetries $N = 2$, hence the supermanifold $\mathbb{R}^{4\vert \mathbf{4}+ \mathbf{4}}$, or rather the super Minkowski spacetime $\mathbb{R}^{3,1\vert \mathbf{4}+ \mathbf{4}}$)
Review includes the following:
I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace
S.J. Gates Jr, M.T. Grisaru, M. Rocek, W. Siegel, Superspace, or One thousand and one lessons in supersymmetry textbook (1983), available as arXiv:hep-th/0108200
Albert Schwarz, On the definition of superspace Teoret. Mat. Fiz., 1984, Volume 60, Number 1, Pages 37–42 (Mi tmf5111), (russian original)