What is the capital of the Vatican?

The Vatican is one out of two independent countries without a capital. The other one is Nauru.

Also, a city state like Singapore has no capital city distinct from the country as a whole, but you might still say that Singapore is the capital of Singapore.
https://en.wikipedia.org/wiki/Capital

I see, though, that this page lists Vatican City as the capital of Vatican City. I am not sure if I agree on that.
https://en.wikipedia.org/wiki/List_of_capitals_by_country

If you look at the similar French Wiki site, you will see that it lists Vatican City with no capital.
http://fr.wikipedia.org/wiki/Liste_des_capitales_du_monde

Vatican City
http://www.nationsencyclopedia.com/Europe/Vatican.html

Many sites claim that Yaren is the capital of Nauru, as does my French reference above. This is wrong. There is no capital. Yaren is a "capital district".

With regards to the Vatican City, I suppose it is a matter of definition. Could a country the size of a golf course, no permanent population and no street adresses have a "capital"? I don't think so. In a case like this, the #1 clue is: Don't believe anything you read in Wikipedia, or anywhere else on the net for that matter. False information will always multiply. My main Geographical reference is - as always - the Merriam Webster's Geographical Dictionary. It does NOT list the Vatican City State with a capital. However, Rome was the capital of the Papal States 754-1870, at least during the periods when the city was actually under the control of the Papacy. I searched the official site of the Vatican City State, and found no reference to a "capital". Besides, one definition of a capital would be that the governmental institutions are placed there. This is mainly not the case in the Vatican.

The Vatican City covers a territory of 44 hectares (roughly 100 acres). It is partly surrounded by walls and stretches into St Peter’s Square as far as a strip of travertine stone that corresponds with the furthest end of the colonnade. This marks the boundary of the State and the edge of the Square which is normally open to everyone. There are five entrances to the Vatican City, each of them guarded by the Pontifical Swiss Guards and by the Gendarmes Corps of Vatican City State.

Because Vatican City is so small, several Departments and offices belonging to the Holy See are situated in buildings around Rome (in Piazza Pio XII, Via della Conciliazione, Piazza San Calisto, Piazza della Cancelleria and in Piazza di Spagna). According to the Lateran Treaty, these buildings enjoy the same status, recognized by international law, as embassies and foreign diplomatic missions abroad. The population of Vatican City is about 800 people, of whom over 450 have Vatican citizenship, while the rest have permission to reside there, either temporarily or permanently, without the benefit of citizenship; The Pope lives in Vatican City.

http://www.travelamalfi.com/tours/991-vatican-city-and-sistine-chapel-tour-skip-the-line?lang=en

But Vatican City is still a nation, a nation that is too small to have a capital city.

http://www.homeworkspot.com/know/smallestnation.htm

The CIA call it The Holy See rather than The Vatican, and list Vatican City as the capital.
https://www.cia.gov/the-world-factbook/countries/holy-see-vatican-city/

For Nauru, they say no capital, but government offices in Yaren.
https://www.cia.gov/the-world-factbook/countries/nauru/

What exactly distinguishes the Yaren district of Nauru, where the seat of government is located, and a capital?
A capital doesn't have to be a major city as shown by Monaco-Ville, the capital of Monaco.
Note also that most reference works cite Amsterdam as the capital of the Netherlands even though the government is in The Hague.
The Vatican City State can be regarded as being coterminous with it's own capital even if the government institutions are elsewhere, or regarded as having no capital at all.
There is no right or wrong here. It just depends on the definition of "capital".

The official website of the "Vatican City State" ("Stato della Città del Vaticano") explains in the section "Stato e Goberno" every single aspect of State and Government issues like legislature, organs and structure of the government, population, flag, hymn, money and stamps and even license plates. However, it does not use the word "capital" not once.

https://www.vaticanstate.va/it/