  Classical differential geometry can be encoded in spectral data, such as
Connes' spectral triples, involving supersymmetry algebras. In this paper, we
formulate non-commutative geometry in terms of supersymmetric spectral data.
This leads to generalizations of Connes' non-commutative spin geometry
encompassing non-commutative Riemannian, symplectic, complex-Hermitian and
(Hyper-)Kaehler geometry. A general framework for non-commutative geometry is
developed from the point of view of supersymmetry and illustrated in terms of
examples. In particular, the non-commutative torus and the non-commutative
3-sphere are studied in some detail.
