We study the space {\small $\nua{m}{d}$} of clouds in {\small $\bbr^d$}
(ordered sets of m points modulo the action of the group of affine
isometries). We show that {\small $\nua{m}{d}$} is a smooth space, stratified
over a certain hyperplane arrangement in {\small $\bbr^m$}. We give an
algorithm to list all the chambers and other strata (this is independent of
d). With the help of a computer, we obtain the list of all the
chambers for {\small $m\leq 9$} and all the strata when {\small $m\leq 8$}. As
the strata are the product of a polygon space with a disk, this gives a
classification of {\small $m$}-gon spaces for {\small $m\leq 9$}. When {\small
$d=2,3\,$}, {\small $m=5,6,7\,$}, and modulo reordering, we show that the
chambers (and so the different generic polygon spaces) are distinguished by the
ring structure of their {\small ${\rm mod}\, 2$}-cohomology.