The 26-year-old Downing, who handed in a transfer request earlier this week, has joined on a long-term deal after passing a medical, the club said on their website.
Doni, who has 10 caps for Brazil, has been brought in as back-up to Liverpool's current number one Pepe Reina.
The Anfield club did not disclose the financial details of either deal but British media had put the transfer fee for Downing, who has 27 caps, at 20 million pounds.
Kenny Dalglish's Liverpool side, who finished sixth last season, have been on a recruitment drive as they bid to break into the top four with Sunderland's Jordan Henderson and Blackpool's Charlie Adam having already joined the club.
"It has been a long time coming and I've had to wait a few weeks, but I'm really pleased to be here," said Downing, who joined Villa from Middlesbrough for 10 million pounds in 2009.
"With the tradition, the manager and the players they have here, there was a big temptation to come here and once I knew of their interest, there was only one place I wanted to go."
Liverpool said the 31-year-old Doni had signed a contract after agreeing personal terms and passing a medical.
The former Brazil international made 179 appearances for Roma in six seasons but has struggled to command a regular starting place at the Stadio Olimpico after falling behind Julio Sergio in the pecking order.
Liverpool Director of Football, Damien Comolli, said the move would help the club's young keepers gain experience.
"We are delighted to sign Doni," Comolli said. "It's a position we felt we had to strengthen because it will allow our young goalkeepers to go on loan and get some experience at some point during the season, which is crucial in their development.
"We are getting a very experienced goalkeeper who is used to handling the pressure of big games and international matches with Brazil. He had a very good end of last season with Roma, so we are very pleased," he added on the club website.
