Multielement taxonomy was instituted for Ordovician conodonts over a decade ago, and probably a majority of the multielement genera have been defined or are well understood. The present systems of notation for elements within apparatuses are inadequate and cumbersome. A new notation scheme is proposed which applies a single-letter code to the position in the apparatus occupied by certain element morphotypes. The taxonomic status of all known Ordovician conodont genera is reviewed (appendix) using the new notation, and a new scheme to classify conodont apparatuses is presented. Five main apparatus types (I-V) and seventeen subtypes (IA-IC, etc.) are defined. Within these groups, all known Ordovician conodont genera can be accommodated, and probably few new groups are required to include all other conodont genera. The apparatus types and subtypes are defined on the basis of symmetry, curvature, and number of the element types, with a clear distinction being made between the first and second transition series. Certain homologous relationships, both between and within many apparatus types, are noted. The evolution of the five major types, and the subtypes, is traced through the Ordovician. The pattern of evolution suggests that the types and subtypes recognized are probably natural biologic groupings, largely reflecting phylogenetic change.