  An association scheme is called quasi-thin if the valency of each its basic
relation is one or two. A quasi-thin scheme is Kleinian if the thin residue of
it forms a Klein group with respect to the relation product. It is proved that
any Kleinian scheme arises from near-pencil on~$3$ points, or affine or
projective plane of order~$2$. The main result is that any non-Kleinian
quasi-thin scheme a) is the two-orbit scheme of a suitable permutation group,
and b) is characterized up to isomorphism by its intersection number array. An
infinite family of Kleinian quasi-thin schemes for which neither a) nor b)
holds is also constructed.
