  The celebrated Zariski Cancellation Problem asks as to when the existence of
an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ for (affine)
algebraic varieties $X$ and $X'$ implies that $X\cong X'$. In this paper we
provide a criterion for cancellation by the affine line (that is, $n=1$) in the
case where $X$ is a normal affine surface admitting an $\mathbb{A}^1$-fibration
$X\to B$ over a smooth affine curve $B$. If $X$ does not admit such an
$\mathbb{A}^1$-fibration then the cancellation by the affine line is known to
hold for $X$ by a result of Bandman and Makar-Limanov. It occurs that for a
smooth $\mathbb{A}^1$-fibered affine surface $X$ over $B$ the cancellation by
an affine line holds if and only if $X\to B$ is a line bundle, and, for a
normal such $X$, if and only if $X\to B$ is a cyclic quotient of a line bundle
(an orbifold line bundle). When the cancellation does not hold for $X$ we
include $X$ in a non-isotrivial deformation family $X_\lambda\to B$,
$\lambda\in\Lambda$, of $\mathbb{A}^1$-fibered surfaces with cylinders
$X_\lambda\times\mathbb{A}^1$ isomorphic over $B$. This gives large families of
examples of non-cancellation for surfaces which extend the known examples
constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a.
