We treat the problem of extending derivations and endomorphisms of a given ring $R$ to a skew polynomial ring $R[x,f,d]$ over $R$. As an application we obtain the general conditions for the existence of such rings in finitely many variables over $R$. We also prove that under suitable conditions, the $d$ (or the $f$)-simplicity of $R$ implies the $f$-simplicity of $R[x,f,d]$.