  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.
