The deepwater boxfishes of the family Aracanidae are the phylogenetic sister group of the shallow-water, generally more tropical boxfishes of the family Ostraciidae. Both families are among the most derived groups of teleosts. All members of both families have armored bodies, the forward 70% of which are enclosed in rigid bony boxes (carapaces). There is substantial intragroup variation in both groups in body shapes, sizes, and ornamentation of the carapaces. Swimming-related morphology, swimming mode, biomechanics, kinematics, and hydrodynamics have been studied in detail in multiple species of the ostraciids. Ostraciids are all relatively high-performance median and paired fin swimmers. They are highly maneuverable. They swim rectilinearly with substantial dynamic stability and efficiency. Aracanids have not been previously studied in these respects. This article describes swimming-related aspects of morphology, swimming modes, biomechanics, and kinematics in two south Australian species (striped cowfish and ornate cowfish) that are possibly representative of the entire group. These species differ morphologically in many respects, both from each other and from ostraciids. There are differences in numbers, sizes, and placements of keels on carapaces. The most important differences from ostraciids are openings in the posterior edges of the carapaces behind the dorsal and anal fins. The bases of those fins in ostraciids are enclosed in bone. The openings in aracanids free the fins and tail to move. As a result, aracanids are body and caudal fin swimmers. Their overall swimming performances are less stable, efficient, and effective. We propose establishing a new category of swimming mode for bony fishes called "aracaniform swimming."