Villas-Boas, who led Porto to a domestic double as well as winning the Europa League last season, replaces Carlo Ancelotti, who was sacked in May after failing to win a trophy in his second season having won the Premier League and FA Cup in his first.
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"Andre was the outstanding candidate for the job," said a statement on the Chelsea website.
"He is one of the most talented young managers in football today and has already achieved much in a relatively short space of time," the statement continued.
"His ambition, drive and determination matches that of Chelsea and we are confident Andre's leadership of the team will result in greater successes in major domestic and European competitions.
"Andre will bring his coaching experience back to a club he is already very familiar with, having previously worked here for three years.
"He has always been highly regarded at Chelsea and everyone here looks forward to welcoming him back and working with him."
The Dutchman, who enjoyed a successful spell as caretaker boss at Chelsea in 2008-09 and has a close relationship with Abramovich from his days as Russia coach, is currently Turkey's national team manager but speculation that he might still arrive as director of football has cooled.
Former Chelsea midfielder Ray Wilkins, sacked as Ancelotti's assistant last November in a move blamed by many observers as contributing to the team's decline through the season, welcomed Villas-Boas's appointment.
"We've had lots of older coaches come in at Stamford Bridge and they've not done as well, so we have to give this young man the credit for what he has done up to now," Wilkins told Sky Sports News.
"Let's give the young man a chance, he's obviously done exceptionally well. His man-management skills from what I can read from the players he has worked with have been first-class.
"That will be required at Stamford Bridge because these are big players, big guys, and it will be very interesting to see how this young man goes on."
Villas-Boas, who proved a shrewd operator in the transfer market, is likely to try to persuade Porto's Colombian striker Falcao to follow him to Chelsea after his prolific season, which included 17 goals in the Europa League alone.
