Kenny – who has led Dundalk to four League of Ireland titles in the last five seasons – will take over Ireland's Under-21 side before moving up to the senior post in two years' time.
McCarthy – who left his role as Ipswich Town boss in April – replaces Martin O'Neill, who stepped down on Wednesday after a difficult year.
They failed to qualify for the 2018 World Cup and then endured a miserable start to life in the Nations League, ultimately getting relegated from Group B4 with just two points and one goal from four matches against Wales and Denmark.
McCarthy will find out next weekend who he will face in his first game in charge when the draw for the Euro 2020 qualifying groups takes place.
CONFIRMED | The FAI are delighted to announce that Mick McCarthy is the new Ireland Manager with Stephen Kenny with taking over as U21 Manager.— FAIreland (@FAIreland) November 25, 2018
Full story: https://t.co/Jwxe1142ow #COYBIG pic.twitter.com/aQdZub628Q
Ireland's record goalscorer Robbie Keane, as well as former Wolves boss Terry Connor, will join McCarthy as assistants.
"I'm honoured and excited to be back with the Republic of Ireland. I am delighted that the FAI Board and CEO have given me this opportunity to lead the team to Euro 2020," said McCarthy.
"I have been involved in club football for the last 15 years, so this will be a new challenge but one that I know well. I'm delighted to have Terry Connor, a trusted colleague and excellent coach, and Robbie Keane, one of the finest players who I had the pleasure of managing, helping me as assistant coaches.
"I am looking forward to getting started and embracing this challenge to qualify for Euro 2020, where it would be fantastic to play in front of the Ireland supporters in Aviva Stadium."