Extending the Stone Duality Theorem, we describe two categories which are dually equivalent to the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. We find as well two categories which are dually equivalent to the category ZComp of all zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces.

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We prove a new duality theorem for the category of precontact algebras which implies the Stone Duality Theorem, its connected version obtained in [16], the recent duality theorems from [3,23], and some new duality theorems for the category of contact algebras and for the category of complete contact algebras.

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