In this article we develop an approach to deformations of the Witt and Virasoro algebras based on $sigma$-derivations. We show that $sigma$-twisted Jacobi type identity holds for generators of such deformations. For the $sigma$-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the $q$-deformations of the Witt and Virasoro algebras associated to $q$-difference operators, providing also corresponding q-deformed Jacobi identities.
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