The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism
X
×
A
n
≅
X
′
×
A
n
X\times \mathbb {A}^n\cong X’\times \mathbb {A}^n
for (affine) algebraic varieties
X
X
and
X
′
X’
implies that
X
≅
X
′
X\cong X’
. In this paper we provide a criterion for cancellation by the affine line (that is,
n
=
1
n=1
) in the case where
X
X
is a normal affine surface admitting an
A
1
\mathbb {A}^1
-fibration
X
→
B
X\to B
with no multiple fiber over a smooth affine curve
B
B
. For two such surfaces
X
→
B
X\to B
and
X
′
→
B
X’\to B
we give a criterion as to when the cylinders
X
×
A
1
X\times \mathbb {A}^1
and
X
′
×
A
1
X’\times \mathbb {A}^1
are isomorphic over
B
B
. The latter criterion is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of
X
X
over
B
B
. It occurs that for a smooth
A
1
\mathbb {A}^1
-fibered surface
X
→
B
X\to B
the cancellation by the affine line holds if and only if
X
→
B
X\to B
is a line bundle, and, for a normal such
X
X
, if and only if
X
→
B
X\to B
is a cyclic quotient of a line bundle (an orbifold line bundle). If
X
X
does not admit any
A
1
\mathbb {A}^1
-fibration over an affine base then the cancellation by the affine line is known to hold for
X
X
by a result of Bandman and Makar-Limanov.
If the cancellation does not hold then
X
X
deforms in a non-isotrivial family of
A
1
\mathbb {A}^1
-fibered surfaces
X
λ
→
B
X_\lambda \to B
with cylinders
X
λ
×
A
1
X_\lambda \times \mathbb {A}^1
isomorphic over
B
B
. We construct such versal deformation families and their coarse moduli spaces provided
B
B
does not admit nonconstant invertible functions. Each of these coarse moduli spaces has infinite number of irreducible components of growing dimensions; each component is an affine variety with quotient singularities. Finally, we analyze from our viewpoint the examples of non-cancellation constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a.