The former West Ham boss has signed a two-year deal at the Stadium of Light and will be tasked with reversing their fortunes
The club narrowly avoided relegation under Dick Advocaat last season and are currently 19th in the Premier League table, having failed to win a single game, which led to the Dutchman's exit last week.
Allardyce has been out of management since leaving the Hammers at the end of last season, and has already expressed his desire to get back into the dugout.
“I have enjoyed my break from football and now I’m raring to get back. I met with Ellis (Short) and we spoke at length about the club and his ambitions and I knew I wanted to be part of that," he told Sunderland's official website.
“I hope to be able to help to bring the stability and success that everyone wants.
“Of course it’s a challenging job, but it’s something I have experience of in the past. I’m looking forward to working with the players and of course I will be relying on the help of the Sunderland supporters, whose tremendous passion I have experienced first-hand.
"I can’t wait to get started.”
Allardyce briefly managed Sunderland's bitter rivals Newcastle, in the 2007-08 season and Short has expressed his delight at the 60-year-old's appointment.
“I am very pleased to welcome Sam to our football club," he added. “Sunderland is a club he knows well and he was the obvious best choice for the job.
"He has vast experience of managing in the Premier League and an understanding first-hand of the north east and the passion of our fans, which will stand him in great stead.
“I would like to assure our fans that once Dick made us aware of his intention to leave, Lee Congerton oversaw an organised and structured recruitment process, that bore very little resemblance to what has been described in the media.
“For example, this was a very popular job, proactively sought after by a large number of managers - contrary to much of what has been portrayed. The process was made easier by the fact that Sam was such an obvious choice.
“The other misconception is that Sam had to be persuaded to join us; nothing could be further from the truth. From the very beginning, he understood the importance of this job and showed great enthusiasm for the role and a desire to be part of moving this club forward.”