  A noncommutative analogue of the Zariski cancellation problem asks whether
$A[x]\cong B[x]$ implies $A\cong B$ when $A$ and $B$ are noncommutative
algebras. We resolve this affirmatively in the case when $A$ is a
noncommutative finitely generated domain over the complex field of
Gelfand-Kirillov dimension two. In addition, we resolve the Zariski
cancellation problem for several classes of Artin-Schelter regular algebras of
higher Gelfand-Kirillov dimension.
