Reexamination of previously described conulariids has shown that the morphology of steeply pyramidal, generally four-sided conulariid exoskeletons is more complex than was realized by earlier generations of conulariid specialists (e.g., Van Iten, 1992a, 1992b; Jerre, 1994; Van Iten et al., 1996). Much of this complexity is expressed as variation in the anatomy of the corners and midlines (Van Iten, 1992a). In nearly all conulariids, the corners are sulcate, and in some genera (e.g., Archaeoconularia Bouček, 1939 and Pseudoconularia Bouček, 1939) the longitudinal center line, or midline, of the faces also is sulcate or is marked by a straight or zigzagged ridge. In the oldest known conulariid, Baccaconularia Hughes, Gunderson, and Weedon, 2000 (Cambrian: Furongian Series), the midlines are marked by a series of elongate invaginations. Corners and/or midlines of many other conulariids exhibit subtle or pronounced internal thickening (e.g., Van Iten, 1992a; Jerre, 1994). Most such conulariids exhibit one or two sets of longitudinal ridges, or carinae, with a single carina at each corner and/or a single carina or a pair of carinae at each midline. The height of the carinae varies considerably between species, ranging from less than one-fiftieth to over one-half the distance to the center of the exoskeletal cavity. Both corner and midline carinae may be continuous or (less frequently) seriated. Also, the abaxial edge of single midline carinae may have a single crest or (rarely) it may be weakly or strongly bifurcate.