Bradley spent seven years as director of rugby at Connacht where he guided the Galway-based side to three Amlin Challenge Cup semi-finals.
He has more recently been involved with the Georgian National side and previously coached Ireland Under-21's and Ireland A.
As an international scrum-half, Bradley helped Ireland win the Triple Crown in 1985, and went on to feature in the 1987 and 1995 Rugby World Cups. In all, he won 40 caps for his country, captaining them 13 times.
Edinburgh chief executive Craig Docherty today welcomed him to the club, saying: "Securing Michael's services as head coach is great news and a real boost ahead of the new campaign.
"His experience of building successful teams and bringing through young talent will stand us in very good stead as we seek to continue to build the club. He is a proven leader, with pedigree in both the international and European arenas."
Bradley said he was "thrilled" to have been given a "massive opportunity", continuing: "I'm honoured and delighted to have been offered the chance to become the head coach at Edinburgh Rugby.
"I'm excited by the challenge of helping the club fulfil its potential, and I can't wait to start working with what is a talented squad. I know a lot of the players from my years in the Magners League with Connacht, and I'm well aware of their capabilities.
"I'm looking forward to spending the summer getting to know the club and the job, so that we're in a position to hit the ground running when the new campaign rolls round."