  A smooth family $\varphi:\mathcal V\to S$ of surfaces will be called {\em
completable} if there is a logarithmic deformation $(\bar {\mathcal
V},{\mathcal D})$ over $S$ so that ${\mathcal V}=\bar{\mathcal V}\backslash
{\mathcal D}$. Two smooth surfaces $V$ and $V'$ are said to be deformations of
each other if there is a completable flat family ${\mathcal V}\to S$ of smooth
surfaces over a connected base so that $V$ and $V'$ are fibers over suitable
points $s,s'\in S$. This relation generates an equivalence relation called {\em
deformation equivalence}. In this paper we give a complete combinatorial
description of this relation in the case of affine ruled surfaces, which by
definition are surfaces that admit an affine ruling $V\to B$ over an affine
base with possibly degenerate fibers. In particular we construct complete
families of such affine ruled surfaces. In a few particular cases we can also
deduce the existence of a coarse moduli space.
