We introduce the fuzzy supersphere as sequence of finite-dimensional, noncommutative Z 2 -graded algebras tending in a suitable limit to a dense subalgebra of the Z 2 -graded algebra of H ∞ -functions on the (2|2)-dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the super-deRham complex are introduced. In particular we reproduce the equality of the super-deRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the "fuzzy level". osp(1|2) [4,42,43], which correspond, when restricted to the even part of osp(1|2), with the compact real form of osp(1|2) 0 . Explicitly they are given by J ‡ λ i := J i J ‡ λ 4 := (−1) λ J 5 (35) J ‡ λ 5 := (−1) λ+1 J 4 .

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We study the graded derivation-based noncommutative differential geometry of the Z 2 -graded algebra M(n͉m) of complex (nϩm)ϫ(nϩm)-matrices with the ''usual block matrix grading'' ͑for n m). Beside the ͑infinite-dimensional͒ algebra of graded forms, the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In particular we prove the universality of the graded derivation-based first-order differential calculus and show that M(n͉m) is a ''noncommutative graded manifold'' in a stricter sense: There is a natural body map and the cohomologies of M(n͉m) and its body coincide ͑as in the case of ordinary graded manifolds͒.

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