Archaea have a single type of RNAP, responsible for the synthesis of all RNA.  Archaeal RNAP is structurally and mechanistically similar to bacterial RNAP and eukaryotic nuclear RNAP I-V, and is especially closely structurally and mechanistically related to eukaryotic nuclear RNAP II.
The history of the discovery of the archaeal RNA polymerase is quite recent. The first analysis of the RNAP of an archaeon was performed in 1971, when the RNAP from the extreme halophile Halobacterium cutirubrum was isolated and purified. Crystal structures of RNAPs from Sulfolobus solfataricus and Sulfolobus shibatae set the total number of identified archaeal subunits at thirteen.
Archaea has the subunit corresponding to Eukaryotic Rpb1 split into two. There is no homolog to eukaryotic Rpb9 (POLR2I) in the S. shibatae complex, although TFS (TFIIS homolog) has been proposed as one based on similarity. There is an additional subunit dubbed Rpo13; together with Rpo5 it occupies a space filled by an insertion found in bacterial β′ subunits (1,377–1,420 in Taq). An earlier, lower-resolution study on S. solfataricus structure did not find Rpo13 and only assigned the space to Rpo5/Rpb5. Rpo3 is notable in that it's an iron–sulfur protein. RNAP I/III subunit AC40 found in some eukaryotes share similar sequences, but does not bind iron. This domain, in either case, serves a structural function.
Archaeal RNAP subunit previously used an "RpoX" nomenclature where each subunit is assigned a letter in a way unrelated to any other systems. In 2009, a new nomenclature based on Eukaryotic Pol II subunit "Rpb" numbering was proposed.