In the present work, we compute quasi-derivations of the Witt algebra and some algebras well related to the Witt algebra. Namely, we prove that each quasi-derivation of the Witt algebra is a sum of a derivation and a \(\frac{1}{2}\) -derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras \(\mathscr {W}(a,b):\) In the case of \(b=-1,\) they do not have interesting examples of quasi-derivations, but the case of \(b\ne -1\) provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of \(\mathscr {W}(a,b).\) As a corollary, we describe the derivations and quasi-derivations of the Novikov–Witt and admissible Novikov–Witt algebras previously constructed by Bai and his co-authors; and \(\delta \) -derivations and transposed \(\delta \) -Poisson structures on cited Lie algebras. In particular, we proved that each \(\mathscr {W}(a,b)\) admits a non-trivial transposed \(\frac{1}{1-b}\) -Poisson structure.
