People might assume that the numbering and listing of popes would be simple and obvious, but that isnt always the case. There have been several names used multiple times over the years: Benedict, Alexander, Boniface, Paul, Clement, Urban, and Gregory to name a few. In two series, though, there are discrepancies and not everyone agrees on who gets what number.
Numbering the Stephens:
There have been either eight or seven popes with the name Stephen, depending upon how you number them. The source of conflict and confusion can be traced back to Stephen II, a popular candidate who was unanimously elected to succeed Zachary in 752.
Unfortunately, Stephen died just a couple of days after his election and before he could be consecrated. Does this mean that he wasnt technically pope? Those who think that a person has to be consecrated, not merely elected, say that Stephen didnt become pope and cant be included in the official lists. This is the dominant view in the Catholic hierarchy. He isnt included in the Annuario Pontificio and doesnt even appear in the Liber Pontificalis.
Stephens successor also took the name Stephen and in the official records is listed as Stephen II. Because of the confusion, though, its far more common to see him listed as Stephen II (III) in recognition of the fact that some records actually describe him as the third Stephen. This site and other places that include both Stephens in the list show him as Stephen III (II).
Numbering the Felixes:
The second series of number discrepancies is found in the Felixes. Three popes have had the name Felix, but the last was named Felix IV. Why? His predecessor is typically listed as Felix III (II) because, at the time of his election (492), an earlier antipope by the name of Felix II had not yet been removed from the list of genuine popes.
Because of this, he had to adopt the name Felix III. Since then the antipope Felix II has been removed so what do we do with Felix III? He has kept that number, but the (II) is add afterward because he is now technically the second pope to have had that name. In 526, another Felix was elected and he is listed as Felix IV (III).

