The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X × A n ≅ X ′ × A n for (affine) algebraic varieties X and X ′ implies that X ≅ X ′. In this and the subsequent papers 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 A 1-fibration X → B over a smooth affine curve B. If X does not admit such an A 1-fibration, then the cancellation by the affine line is known to hold for X by a result of Bandman and Makar-Limanov. We show that, for a smooth A 1-fibered affine surface X over B, the cancellation by an affine line holds if and only if X → B is a line bundle, and, for a normal such X, if and only if X → B is a cyclic quotient of a line bundle (an orbifold line bundle). When cancellation does not hold for X, we include X in a non-isotrivial deformation family X λ → B, λ ∈ Λ, of A 1-fibered surfaces with cylinders X λ × 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. Given two A 1-fibered surfaces with reduced fibers and the same Danielewski-Fieseler quotient ˘ B → B, we provide a criterion as to when the corresponding cylinders are isomorphic over B. This criterion is expressed in terms of linear equivalence of certain 'type divisors' on ˘ B. We construct a coarse moduli space of A 1-fibered surfaces with no multiple fiber and with a given cylinder provided the base curve does not admit any nonconstant invertible function.