Cardinal Keith O'Brien apologised to abuse victims in his address
Catholic leaders in Britain and Ireland have used their Easter sermons to address the Church's handling of its global child abuse scandal.
The leader of Catholics in England and Wales, Archbishop Vincent Nichols, said "serious sins" had been committed.
The Irish Church's leader Cardinal Sean Brady - who has faced resignation calls - said there was no longer a "hiding place for abusers in the church".
Meanwhile the Archbishop of Canterbury has apologised over previous remarks.
Dr Rowans Williams, leader of the Anglican Communion, had said in an interview that Ireland's Catholic Church was losing all credibility.
He said in a subsequent BBC interview that he was "sorry" for adding to the difficulties being faced by Irish bishops.
He did not tackle the child abuse issue in his own sermon on Sunday.
At the Vatican, Pope Benedict XVI made no direct mention of the cover-up accusations which have engulfed the Church.
He said Easter brought a message of pardon, goodness and truth.
At St Mary's Cathedral in Edinburgh, Cardinal Keith O'Brien, head of the Catholic Church in Scotland, apologised to abuse victims.
He said: "Crimes against children have indeed been committed, and any Catholics who were aware of such crimes and did not act to report them brings shame on us all."
He added that "no comfort" could come from the fact that only a small percentage of priests were guilty of paedophilia.
'Need for forgiveness'
Meanwhile Archbishop Nichols told worshippers at Westminster Cathedral that to appreciate the Easter message "we have to begin with our own sin and shame".
He said: "In recent weeks the serious sins committed within the Catholic community have been much talked about.
"For our part, we have been reflecting on them deeply, acknowledging our guilt and our need for forgiveness."
The head of Ireland's Catholics, who has been facing pressure to resign for his role in mishandling the case of a serial child abuser, admitted to worshippers that he was part of a cover-up culture.
Cardinal Sean Brady: ''My overriding concern will always be the safety and protection of everyone in the Church, especially children''
Speaking on Easter Sunday, Cardinal Sean Brady told the congregation at St Patrick's Cathedral in Armagh the Pope had talked of a "misplaced" concern for the Church's reputation.
He said: "I realise that, however unintentionally, however unknowingly, I too allowed myself to be influenced by that culture in our Church, and our society.
"I pledge to you that, from now on, my overriding concern will always be the safety and protection of everyone in the Church - but especially children and all those who are vulnerable."
Last month Cardinal Brady admitted he represented the Church at meetings in 1975 where children signed vows of silence over complaints against paedophile priest Fr Brendan Smyth.
The global Catholic Church has been rocked by sex abuse scandals - many dating back decades - in Switzerland, the Netherlands, Austria, Germany, the US and Ireland.
Two major reports into allegations of paedophilia among Irish clergy last year revealed the extent of abuse, cover-ups and hierarchical failings.
Dr Williams' comments about the controversy had angered the Catholic Archbishop of Dublin, Diarmuid Martin, who said he was "stunned" by them.
Dr Williams later telephoned the archbishop to express his "deep sorrow" and insist he meant no offence.
Dr Rowan Williams: "Our future is bound up with whether we are able to turn to those we have hurt and seek forgiveness"
Speaking on Aled Jones' BBC Radio 2 show on Sunday, Dr Williams said he did not think he had said anything that had not already been said by others, including the leaders of the Irish Church.
At a Mass in Rome a leading cardinal, Angelo Sodano, said Catholics would not be influenced by what he called the "petty gossip" of the moment.
Meanwhile, the Pope's personal preacher has apologised for comments in which he likened criticism of the Roman Catholic church over child sex abuse to the persecution of Jews.
Last month, the Pope apologised to all victims of child sex abuse by Catholic priests in Ireland.
He has also rebuked Irish bishops for "grave errors of judgement" in dealing with the problem.