The definitions of the category of finite Witt-correspondences and of a presheaf with Witt-transfers are given. The injectivity on the affine line, the excision isomorphism on the affine line, and the excision isomorphism for an étale morphism of curves are proved. The homotopy invariance of the Nisnevich sheafification Fnis of a homotopy invariant presheave with Witt-transfers F is proved, and the Nisnevich cohomologies H_nis ⁱ (U,Fnis) are shown to be trivial for any U ⊃ A¹ and i > 0.