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We introduce a class of non-commutative geometries, loosely referred to aspara-spaces, which are manifolds equipped with sheaves of non-commutativealgebras called para-algebras. A differential analysis on para-spaces isinvestigated, which is reminiscent of that on super manifolds and can bereadily applied to model physical problems, for example, by using para-spaceanalogues of differential equations. Two families of examples, the affinepara-spaces $\mathbb{K}^{m|n}(p)$ and para-projective spaces$\mathbb{KP}^{m|n}(p)$, with $\mathbb{K}$ being $\mathbb{R}$ and $\mathbb{C}$,are treated in detail for all positive integers $p$. As an application of suchnon-commutative geometries, we interpret Green's theory of parafermions interms of para-spaces on a point. Other potential applications in quantum fieldtheory are also commented upon.

Para-spaces, their differential analysis and an application to Green's quantisation