Secret archives at the Vatican and in each diocese worldwide
The Vatican points out that the Latin word doesn't necessarily mean "secret — yet it runs its archives as if it did. For more on the Church's troubled relations with the historical record see:
♦ Pope’s claim to temporal power based on 8th-c forgery
♦ Timely eviction of the Slovak National Memory Institute (2007)
♦ The Vatican and the Holocaust: A Preliminary Report International Catholic-Jewish Historical Commission (2000)
Secret Archives at the Vatican
All concordat originals — about 200 of them — are kept in the Vatican Secret Archives. It holds documents spanning about 1200 years, (8th to 20th centuries). Its shelves and reading rooms are spread over several buildings, including a large bunker beneath the Courtyard of the Pine Cone (Cortile della Pigna, marked by an “X”. The courtyard was named for its gigantic gilt bronze Roman fountain which has now been moved to the Museum.) Under the huge lawn lies a a fire-proof, two-storey reinforced concrete vault containing half of the collection of the 52 miles of shelves in the entire archives. 
Here the Vatican claims to hold every concordat except for the first one, the 1107 Concordat of London. Apparently they lost it and now they don't want to count it, despite the fact that they used to acknowledge this as the first concordat.  It remains, however, in the form of a summary written out by what one would think they should count as a reliable witness — the priest who served as the private secretary of the saint who negotiated it. This very first concordat was tossed out by Britain at the time of the Reformation — a precedent they may not wish to emphasise.
The Vatican archives have many layers of secrecy
Where the Vatican can, it now tries to expunge the self-description "secret". The "oath of secrecy" involved in Church trials (in Latin, iuramentum de secreto servando, literally, "oath to preserve secrecy") is now referred to as the "oath of confidentiality".  However, it can't very well rename the archives which are widely known under their Latin name, Archivum Secretum Apostolicum Vaticanum. So it can only claim that "secretum" doesn't actually mean that it's "secret", but rather that it is the "private" archive of the pope, "over which he exercised supreme and sole jurisdiction". 
This is a bit unconvincing, since it's clear that this is a private archive, not one set up by the government of a real state. But this particular private archive differs profoundly from most other ones, such as the archives set up by companies, societies, institutions and individuals. And the main direction in which it differs is in its degree of secrecy.
♦ The Vatican archives are run by a cardinal who, like other cardinals, has sworn to preserve the secrets of the Church. (This is done in a roundabout fashion. Can 833.2 says that cardinals must make a Profession of Faith, which states that they must observe canon law and this, in turn, contains more than two dozen warnings against causing "scandal" for the Church.)
♦ All documents are normally sealed for 75 years.
♦ The Vatican archives lack a complete catalogue, and even “publication of the [partial] indexes, in part or as a whole, is forbidden”. 
♦ Scholars have been allowed in the archive since 2003, so long as they know exactly which document they’d like a look at — browsing is not allowed. 
♦ Only approved scholars are admitted and even these have to present their research requests in writing in advance, thus, as a cynic remarks,
allowing the librarians ample time to decide between their three options in responding — 1) bring out the requested document, 2) claim the document doesn't exist, or 3) admit the document exists but refuse to give the scholar access to research specific topics. 
♦ Noted scholars given a special invitation have been denied access when they tried to follow this up in a professional manner. When members of the group invited to clear the name of the wartime pope, Pius XII asked to look at the evidence in the Archives, they were told that this request showed that they were anti-Catholic. See: The Vatican and the Holocaust: A Preliminary Report by the International Catholic-Jewish Historical Commission (2000)
♦ By contrast, they made a great show of allowing free access to a popular author wanting to do a coffee-table book on the Secret Archives — someone whose lack of historical knowledge presumably kept him from posing much of a threat. However, even he wasn’t allowed to look at any documents that dated from after 1939. The reason given was that these include papal annulments of marriages of people who might still be alive. This ensured that he didn't stumble on anything about the conduct of Pius XII during the Second World War....
♦ And there is more. A scholar who has worked for years in the Archives asks,
What action is taken by a scriptor, custodian, or prefect when, in the course of his work, he comes across material that is morally or theologically controversial? [...] Such documents may be omitted from the inventories, bound in volumes containing documents of a very different kind, and relegated to some fondo that is closed because of chronological limitation or very seldom consulted. This happened with the personal letters of Pope Borgia to the little clan of his devoted women, and with the original summary of the process of Giordano Bruno, and may have happened many other times that we do not know about. 
Two examples that we do know about came to light only recenlty, when scholars stumbled on embarassing documents which had been "misplaced". In 2001 it was found that a key document about the Templars who were charged with heresy and killed off in the 14th century had been slipped in with unrelated documents, so that it had remained unknown for 700 years.  And a few years later a leading scholar of the Catholic Church found another "misplaced" document, one that dealt with sex and heresy in a 19th-century convent in Rome. 
The Vatican Secret Archives, with their curiously "misplaced" documents and their controlled access seem designed to produce conclusions favourable to the Church. A scholar's access can be revoked at any time, which puts pressure on him not to write anything the Vatican might find seriously damaging. In this conection see Vatican anti-Judaism versus Nazi anti-Semitism: a subtle theological distinction.
Secret archives in every diocese
Even more secret than the archives of the pope are the ones held worldwide by every bishop. This is anchored in Church law. Canons 486-490 stipulate that each diocese must have both a locked archive and a totally secret one, as well. Only the bishop holds the key to this closely guarded secret archive for “documents of criminal cases in matters of morals”.
In 2012 the secret archives of the diocese of Cloyne in Ireland revealed "stunning data — recorded in meticulous detail — on abuse allegations dating over several years, related correspondence, clerical responses and the diocesan handling of the individual complaints." The police had no idea what had been going on. 
In the US the struggle to open some of these diocesan archives came earlier and has been more successful. In 2002, after the Archdiocese of Philadelphia refused to cooperate in an investigation of its handling of sex-abuse claims, a grand jury subpoenaed the church's internal records. There the investigators discovered that the locks and keys mentioned in canon law were supplemented with “alarms, safes and computer security programs”. 
In Philadelphia, the sole keyholders were the cardinal and his closest aides. The files were kept in a row of unlabeled, gray-green cabinets in a windowless room on the 12th floor of the archdiocese’s Center City office tower. Inside was an exhaustive compendium of scandals dating back more than 50 years: priests with drinking problems, priests who had gotten women pregnant, aging stacks of confiscated pornography. Then there were the reams of carefully typed memos that discussed priests with what the archdiocese delicately referred to as “unnatural involvements” or “unusual patterns.” Priests, in other words, who had sexually abused the children in their care....[The] prosecutors were stunned to find thousands of documents that detailed the hundreds of victims who had allegedly been abused by 169 priests. 
Another astoiunding find was made in 2015 when a search warrant gave the police in Pennsylvania access to the secret archive of the Altoona-Johnstown Diocese. There they found evidence that the diocese knew of much of the sexual abuse of hundreds of children by over 50 priests or religious leaders, yet had covered it up for 40 years. This was long enough to have allowed some of the abusers to die without ever facing justice and also long enough to let the statute of limitations expire. But it still wasn't long enough for some of the traumatised victims to be able to face laying charges. The bishop's archives had held their secrets long enough that no criminal charges could be laid. 
These files survived presumably because the Church did not try to punish canon law provides for their elimination once they are no longer needed by the Church. After “the accused parties have died or ten years have elapsed from the condemnatory sentence” of a secret Church court these documents “are to be destroyed” (Canon 489 §2). Only a summary and the text of the Church judgement is to be retained — and, of course, these are still to be kept under the heaviest security.
Even when evidence has not been destroyed, as laid down in canon law, the Church can make documents simply disappear when the need arises. In 2010 when widespread clerical abuse was finally uncovered in Germany, public pressure led to demands for access to the secret archive of the Archdiocese of Munich and Freising. This was the hidden store of documents overseen by Archbishop Ratzinger from 1977 to 1982, before he became pope. But when the lawyer appointed to investigate examined the archives, he found that the documents for the relevant period were missing. “We are dealing with the extensive destruction of files”, he said, a “systematic system of cover-up”. 
Some of the contents of secret diocesan archives can now be found online. Mostly in the US, hundreds of thousands of secret church documents have been obtained in the course of lawsuits and, despite Church opposition, unsealed. BishopAccountability, an online archive established by lay Catholics,  has posted a large number of these diocesan and religious order documents. 
1. Archivum Secretum Vaticanum: "Bunker and storerooms". http://www.archiviosegretovaticano.va/en/archivio/ambienti/bunker-e-depositi/
2, "This may be called the first concordat, unless the agreement of London (1107) is reckoned, as it may be, among the number of concordats." "Concordat", Catholic Encyclopedia, 1908.http://www.newadvent.org/cathen/04196a.htm
3. Commission of Investigation, Report into the Catholic Archdiocese of Dublin, July 2009; released November 26, 2009, "Confidentiality", 4.82. http://www.bishop-accountability.org/reports/2009_11_26_Murphy_Report/04_The_Role_of_Canon_Law.pdf
4. Archivum Secretum Vaticanum: "The Archives". http://www.archiviosegretovaticano.va/en/archivio/
5. Vatican Secret Archives, "Rules for scholars", no. 16. http://asv.vatican.va/en/fond/amm.htm
6. "The Vatican Archive: the Pope's private library", Telegraph, 1 June 2010. http://www.telegraph.co.uk/culture/books/7772052/The-Vatican-Archive-the-Popes-private-library.html
7. "Secret Archives of the Vatican". http://www.rotten.com/library/religion/secret-archives-vatican/
8. "The Vatican Archive: the Pope's private library", Telegraph, 1 June 2010. http://www.telegraph.co.uk/culture/books/7772052/The-Vatican-Archive-the-Popes-private-library.html
9. Maria Luisa Ambrosini with Mary Willis, The Secret Archives of the Vatican, (New York: Barnes & Noble, 1969), p. 303.
10. Gerard DeGroot, Book review: “The Nuns of Sant’Ambrogio” by Hubert Wolf, Washington Post, 5 March 2015. http://www.washingtonpost.com/opinions/a-false-saint-and-sexual-predator-in-a-warped-rome-convent/2015/03/05/072069da-b14d-11e4-827f-93f454140e2b_story.html
11. Philip Pullella, "Knights Templar Win Heresy Reprieve After 700 Years", Reuters, 12 October 2007. http://www.reuters.com/article/idUSL093422320071012
After the document was discovered by a researcher, the Vatican put its own interpretation on events, complaining that poor Pope Clement V had been bullied by the French king into persecuting the Templars. At the same time it offered copies of a collection of these documents for $8,377.
"Vatican to tell true knights' tale: Publishes Templar trial documents", Associated Press, 13 October 2007. http://www.boston.com/news/world/articles/2007/10/13/vatican_to_tell_true_knights_tale/
12. "Stunning flood of claims held in secret diocesan archive", Irish Independent, 02 December 2012. http://www.independent.ie/irish-news/stunning-flood-of-claims-held-in-secret-diocesan-archive-26644556.html
13. “PA priest-abuse files kept behind lock, key, alarm”, Associated Press, 17 May 2012. http://www.google.com/hostednews/ap/article/ALeqM5jmb6WhQjxUFgeiuPv2he71zhcE1g?docId=de4231123c1247e39c04a06ec859ba4e
14. Sabrina Rubin Erdely, “The Catholic Church’s Secret Sex-Crime Files”, Rolling Stone Magazine, 15 September 2011. http://www.rollingstone.com/culture/news/the-catholic-churchs-secret-sex-crime-files-20110906
15. "Grand jury: 2 bishops hid sex abuse of hundreds of children", AP, 1 March 2016. http://bigstory.ap.org/article/538b2019e2f648059e0cebe2fab073d0/grand-jury-2-bishops-hid-sex-abuse-hundreds-children
"Pennsylvania Diocese Leaders Knew of Sex Abuse for Decades, Grand Jury Says", New York Times, 1 March 2016. http://www.nytimes.com/2016/03/02/us/pennsylvania-diocese-leaders-knew-of-sex-abuse-for-decades-grand-jury-says.html
16. “German study finds systematic cover-ups in Catholic priest abuse cases”, Deutsche Welle, 3 December 2010. http://www.dw-world.de/dw/article/0,,6294220,00.html
17. "Sex-abuse crisis is a watershed in the Roman Catholic Church's history in America", Philadelphia Inquirer, 25 June 2012. http://articles.philly.com/2012-06-25/news/32394491_1_canon-lawyer-catholic-priests-catholic-bishops
18. "Introduction to the Archives", http://bishopaccountability.org/Introduction_to_the_Archives/