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We propose a definition of Leibniz cohomology, $H{L}^{*}$, for differentiable manifolds. Then $H{L}^{*}$ becomes a non-commutative version of Gelfand-Fuks cohomology. The calculations of $H{L}^{*}({\mathbf{R}}^n;\mathbf{R})$ reduce to those of formal vector fields, and can be identified with certain invariants of foliations.