Originally described by Wagner (1853) as Ichthyosaurus leptospondylus, it has had an unstable taxonomic history.
It was closer in form to Cymbospondylus and Mixosaurus than more advanced genera like Ichthyosaurus: it did not have the dolphin-like form of later ichthyosaurs; it had a more lizard-like appearance.
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).
There are now more than 1,000 people claiming to have witnessed the beast which is reported to measure around fifteen meters long with spikes on its back and appears similar to a Plesiosaur or Ichthyosaurus.
The name means "Mixed Lizard", and was chosen because it appears to have been a transitional form between the eel-shaped Ichthyosaurs such as Cymbospondylus and the later dolphin-shaped ichthyosaurs, such as Ichthyosaurus.
The skull is 860 mm long, but is unusual in having a very short rostrum (~0.6× the skull length), similar to Ichthyosaurus breviceps.
Parvipelvia is a node-based taxon defined in 1999 as "the last common ancestor of Hudsonelpidia, Macgowania, Ichthyosaurus and all of its descendants".
The remains from Australia were originally called Ichthyosaurus australis.
It was so-named ("near lizard") by William Conybeare and Henry De la Beche, to indicate that it was more like a normal reptile than Ichthyosaurus, which had been found in the same rock strata just a few years earlier.
Another thunnosaur described from the same locality, by Bogolubow (1909), is "Ichthyosaurus" steleodon.
Named by Ryosuke Motani, in 1999, it contains the basal taxa like Ichthyosaurus and Stenopterygius and the family Ophthalmosauridae.
It has also been named Ichthyosaurus trigonus posthumus, and identified with Macropterygius and Brachypterygius extremus.
Many specimens have been referred to Ichthyosaurus trigonus; Bauer (1898) suggested that – what are now – Ophthalmosaurus, Brachypterygius and Nannopterygius should be synonymised into Ichthyosaurus trigonus, as this is the older name.