Published Wednesday, 11 September 2013
It comes the day after the Republic's bid to reach the 2014 World Cup in Brazil was effectively ended with a disappointing 1-0 defeat against Austria.
Former Sunderland manager and Northern Ireland international Martin O'Neill is the favourite to succeed Trapattoni.
The FAI held talks with Italian on Wednesday morning, during which it was agreed that he and assistant Marco Tardelli would leave by mutual consent.
Speaking afterwards, Trapattoni said: "I want to thank everyone in Ireland who has given us their support during our time here which has always meant a lot to us.
"We leave this country with emotion because we understand the Irish supporters who have a well-deserved international reputation and they have our utmost respect."
The 74-year-old had previously accepted that his time as Republic boss was coming to an end after calls for him to go in the wake of Friday's 2-1 home defeat by Sweden, which left their qualification hopes in tatters. His contract was due to expire next summer.
Trapattoni was appointed to the Republic job in February 2008.
During his time in charge he guided the team to qualification for the Euro 2012 with a play-off win over Estonia - their first major tournament in 10 years.
He also presided over the World Cup 2010 play-off loss to France, which was decided after Thierry Henry's handball controversy.
Paying tribute, FAI Chief Executive John Delaney said: "We thank Giovanni Trapattoni, Marco Tardelli and Fausto Rossi for the last five and a half years.
"This particular World Cup campaign has been disappointing but Giovanni leaves us with a group of good young players which should form the basis of the squad that the new manager will use for the European Championships in France 2016 when 24 teams qualify."
Bookmakers have tipped former NI captain O'Neill, who hasn't worked since leaving his post at Sunderland, as the leading candidate to replace Trapattoni
Leeds manager Brian McDermott, former Ireland manager Mick McCarthy and former Ireland captain Roy Keane are also among the frontrunners.
The board of the FAI said it will meet in due course to discuss the appointment.
© UTV News