Cardinal Brady, who turns 75 tomorrow, has faced repeated criticism over his role in dealing with allegations of child sex abuse in the Catholic Church.
In particular there have been widespread calls for his resignation after his role in investigating serial abuser Fr Brendan Smyth became known.
In 2010 it emerged he asked those involved in the internal church probe into Brendan Smyth to maintain confidentiality.
Brady later spoke about how Irish people have lost their capacity for mercy and forgiveness.
In a statement, he says his coadjutor Archbishop Eamon Martin will replace him as head of the Catholic Church here as soon as Pope Francis accepts his resignation.
