Named by Ryosuke Motani, in 1999, it contains the basal taxa like Ichthyosaurus and Stenopterygius and the family Ophthalmosauridae.
Both the original description by Maisch and the redescription of the English specimens found that Hauffiopteryx might be either a basalmost member of Eurhinosauria or a basalmost member of Thunnosauria (which is an equivalent position to a basalmost member of Stenopterygiidae sensu Maisch 2008 with exclusion of Ichthyosaurus).