The former Archbishop of Dublin was appointed Cardinal in 2001
Cardinal Desmond Connell has died at the age of 90.
Archbishop of Dublin Diarmuid Martin said the Cardinal had been ill for some time.
He 'passed away peacefully during the night in his sleep'.
Desmond Connell was born in 1926. He served as Archbishop of Dublin from 1988 until 2004 before becoming Cardinal in 2001.
Funeral arrangements will be announced later.
In a statement, a spokesperson for the Taoiseach said: "Cardinal Connell had a long and distinguished academic career and while his time as Archbishop was controversial, those who knew him recognised his desire for holiness and his gentleness and humility of character. The Taoiseach offered his sincere condolences to his family."
Fianna Fáil's Michéal Martin also passed on his condolences, saying: “Dr Connell was a strong advocate for social justice and highlighted the difficulties facing communities in inner-city Dublin throughout his ministry. Throughout his life he dedicated great passion and commitment to the people of his diocese.
“Dr Connell led the Dublin diocese at a very difficult time and recognised the need to allow younger generations to take-up leadership roles in the Church," he added.
President Higgins has also expressed his condolences to Cardinal Connell's family and colleagues.
Cardinal Connell was criticised for his handling of clerical sex abuse cases, and at one point issued a statement asking for forgiveness of those who suffered "unspeakable abuse".
Fr Tony Flannery from the Association of Catholic Priests says the Cardinal - who he described as a "very spiritual man" - found the situation very difficult.
"He was a Church person in every fibre of his being," Fr Flannery told the Pat Kenny Show.
"If he had become Archbishop of Dublin at an earlier stage, it could have worked out for him.
"But then he had the misfortune that he was there when the clerical sex abuse abuse scandal broke, and it was completely beyond his ability to handle."
Fr Flannery added: "The other thing he did, which only emerged later, is he did challenge the Vatican over their failure to respond to the sex abuse scandal [...] He too was let down by the Vatican, and that's another part of the sadness of his life.
"At this stage all we can do is hope and wish that he finds peace in the next life that I presume he did not find in this life."