  In this work we study the automorphisms of skew $PBW$ extensions and skew
quantum polynomials. We use Artamonov's works as reference for getting the
principal results about automorphisms for generic skew $PBW$ extensions and
generic skew quantum polynomials. In general, if we have an endomorphism on a
generic skew $PBW$ extension and there are some $x_i,x_j,x_u$ such that the
endomorphism is not zero on this elements and the principal coefficients are
invertible, then endomorphism act over $x_i$ as $a_ix_i$ for some $a_i$ in the
ring of coefficients. Of course, this is valid for quantum polynomial rings,
with $r=0$, as such Artamonov shows in his work. We use this result for giving
some more general results for skew $PBW$ extensions, using filtred-graded
techniques. Finally, we use localization for characterize some class the
endomorphisms and automorphisms for skew $PBW$ extensions and skew quantum
polynomials over Ore domains.
