Some priests, monks and nuns engage in what would normally be considered healthy relations with consenting adults, however if caught, they are typically defrocked.
Sadly there are worse cases, far too often Roman Catholic priests have been caught with their trousers down abusing kids or adolescents, other times Roman Catholic priests even lift their cassocks to abuse an altar boy for example, just before or just after a religious service. See Roman Catholic Mass and Priestly Child Abuse for more about this outrage.
Victims of priestly paedophilia sometimes suffer lifelong psychological damage.
These perpetrators of a terrible crime tend to keep their jobs, and their bosses who played a part in the cover up get sinecures at the Vatican as a reward, Cardinal Bernard Francis Law of the Boston Archdiocese, specifically. 
In the early 2000s a scandal broke that has rippled worldwide, involving Roman Catholic priests engaging in inappropriate activities with children who they had access to. Many cases were not prosecutable, as the evidence uncovered was from so long ago that the Statute of Limitations had expired. As a result of some of the cases, there was a push to alter these statutes so that molested children would have a chance, as adults, to testify and seek justice. 
Even in the Roman Catholic Church these are small minorities.
Blaming the devil
Inevitably a few in the clerical hierarchy defended its interests by comparing criticism of it to a struggle between unseen forces. According to exorcist Father Gabriele Amorth, none other than Satan is behind the criticism: "There is no doubt about it. Because Benedict XVI is a marvelous Pope and worthy successor to John Paul II, it is clear that the devil wants to ‘grab hold’ of him." What is more, Satan makes use of priests to delegitimize the entire Church: "The devil wants the death of the Church because she is the mother of all the saints". Thus in Fr. Amorth's moral universe the church is never ultimately responsible for the actions of its agents. One obvious question to ask is "why should any thinking person accept moral instruction from people with such poor moral sensibilities?". As further evidence of the hierarchy's obtuseness, the Vatican issued revised internal church rules on July 15 2010, that declared both child sexual abuse and The ordination of women to be "grave moral defects".
Blaming the disease
Cardinal Wilfrid Fox Napier, of South Africa describes paedophilia as "illness, not a criminal condition" and claims abusers should be treated rather than punished. The cardinal says doctors should examine child abusers.  Treatment as well as punishment is common in liberal countries for illnesses that lead to crime.
This looks like a case where protecting brother priests becomes more important than protecting victims. Cardinal Napier recognises that child sex abuse is damaging because he cites abusive priest who were themselves abused during childhood. he claims those priests were damaged and therefore unfit to stand trial. This leaves open the question why the Roman Catholic Church did not spot that these men were so badly damaged. Alternatively one has to ask why the Church allowed men who are not responsible for their actions to become priests. After citing those priests Napier seemed to suggest changing the law so active paedophiles would not be punished.  Naturally fewer potential paedophiles would deterred under such a new law and more children would be at risk of harm. B Dorries from Chicago based Survivors Network of those Abused by Priests and, herself a victim of priestly abuse when she was a child told the BBC:
|“||"If it is a disease that's fine, but it's also a crime and crimes are punished, criminals are held accountable for what they did and what they do. (…) "The bishops and the cardinals have gone to great lengths to cover these crimes to enable the predators to move on, to not be arrested, to keep the secrets within the church."||”|
Napier later apologized unreservedly. 
PZ Myers highlighted Cardinal Roger Mahony who repeatedly concealed crimes by priests against children meriting jail sentences. Myers noted how particular types of vulnerable children were targeted. Myers quotes another source, probably the New York Times.
“Sounds good — please proceed!” the cardinal, now retired, instructed in 1987 after the aide, Msgr. Thomas Curry, cautioned against therapy for one confessed predator — lest the therapist feel obliged to tell authorities and scandalize the archdiocese. The two discussed another priest, Msgr. Peter Garcia, who admitted specializing in the rape of Latino immigrant children and threatened at least one boy with deportation if he complained. 
An ordinary contributor added the following comment
Look, it’s all perfectly consistent, and not at all about keeping the Catholic Church in power. It’s really about the hierarchy’s concern for the Catholic laity, for the Little People in the pews who need to believe in God. If they doubt their church, then they doubt God. Learning about pedophile priests is not faith-promoting. Case closed. Sophisticated theology never forgets that the unsophisticated believer relies more on a simple faith. No need to get all complicated on them by introducing unnecessary crises of faith. (This is comment no 7. )
Abusers and enablers
It is estimated that two percent of Roman Catholic priests were/are abusers and Monks abuse as well. Many of the remaining 98% are enablers, they cover up abuse when they know or suspect it and allow abuse to continue. Non-Catholics can speculate on what “education” turns the devout but naive young men who enter Seminaries into mature priests willing to help abusers indirectly. Abusers are frequently acting under a compulsion, therefore though they are very dangerous people they may be not fully responsible for their actions, enablers do not have this defense though they have the different defense that the Roman Catholic Church has brainwashed them.
- The US Priest Scandal: Romeâ€™s Strategy for Damage Limitation
- "Satan behind media attacks on the Pope, asserts Italian exorcist." Catholic News Agency. 2010 March 31.
- Rachel Donadio. "Vatican Revises Abuse Process, But Causes Stir", The New York Times, 2010 July 16.
- It is an illness but so are many conditions that the law deals with like homicidal mania, compulsive stealing (kleptomania) drug addiction when the addict continually commits crimes to feed the habit, alcoholism when the alcoholic continually breaks the law after drinking.
- Since the firestorm of protest started Napier tried to backtrack, he now says he was quoted out of context. (If this is true it is totally out of character in terms of what the British public expects from the BBC.) Napier now seems to claim that paedophiles are so damaged by sex abuse done to them in childhood that they collectively are unfit to plead. Anger over Cardinal’s gaffe. (It would certainly benefit the Roman Catholic Church if paedophile priests were quietly shunted off to psychiatric hospitals without their crimes being made public in court.) In most cases child sex abuse does not damage a person so badly that person during adulthood is not responsible for his/her actions. Surveys suggest 10% of the population experienced some type of child sex abuse and clearly 10% of the population are not so badly damaged that lose responsibility for their actions. A small minority of paedophiles are probably so badly damaged by what was done to them in childhood that they are not fit to stand trial and belong in psychiatric hospitals. Still it is not reasonable to generalise from that to the whole paedophile population. Law enforcement agencies in all countries with experience of paedophiles are convinced the majority are responsible for their actions and fit to stand trial, this applies even in liberal countries.
- 'Paedophilia not criminal condition' says Durban cardinalPaedophilia 'not a criminal condition', says leading Catholic
- South African cardinal apologizes for saying paedophilia not a crime
- Sophisticated theology
- Does Catholic celibacy contribute to child sex abuse?
- Sexual abuse was 'endemic' in Catholic run Irish boys' homes
- Thousands beaten, raped in Irish reform schools More on abuse by the Roman Catholic Church in Ireland
- 'Everywhere I turned, I ran into sexual abuse' This priest was employed specifically to work in parishes where an abusive priest had been moved out and calm things.
Monks do it as well
- Monk jailed for school sex attacks: Principal of Catholic home given seven years for assaulting boys
- Catholic monk found guilty of Rickmansworth attacks
- Those old fat Catholics This video is about priestly child sex abuse worldwide.
Adapted from Liberapedia where I wrote a large part but not all and incorporating RationalWiki material