Alisson kept 17 clean sheets in 37 Serie A appearances for Roma last season and helped the club reach the Champions League semis and started all five games for Brazil at the World Cup.
Liverpool signed Brazil international Alisson from AS Roma on Thursday, smashing the world record for a goalkeeper in a deal worth up to 72.5 million euros ($84 million).
The Serie A side confirmed in a statement the fee that Liverpool had agreed to pay for the 25-year-old, which eclipses the previous record of 53 million euros paid by Juventus for Gianluigi Buffon in 2001.
"I'm really happy, it's a dream come true to wear such a prestigious shirt for a club of this size that is used to always winning," said Alisson.
"In terms of my life and my career, it's a huge step for me being part of this club and this family."
Liverpool manager Juergen Klopp said the opportunity to sign "one of the world’s best goalkeepers" was too good to turn down.
“At one point in the last few weeks it came up, the opportunity to sign one of the world’s best goalkeepers - then it’s not a long thought, to be honest, it’s only that you need to have a little talk with the owners! They were quite excited, so we did it.
“I think it’s something we have to do. He has nothing to do with the price, we have nothing to do with the price, it’s the market, that’s how it is and we will not think a lot about it."
Roma's sporting director Monchi had earlier said it was a difficult decision for the Italian club, who sold Mohamed Salah to Liverpool last year, to part with another of their prized assets.
"It's hard when you're a sporting director. We received a significant offer for Alisson and we evaluated it," Monchi told reporters.
"In the end, we took a decision and, speaking with Liverpool, we reached an agreement. It's not that we lack ambition. We've only sold two players and we've signed 10, and we're going to continue looking for players."
Alisson kept 17 clean sheets in 37 Serie A appearances for Roma last season and helped the club reach the Champions League semi-finals, where they were knocked out by Liverpool.
He started all five games for Brazil at the World Cup in Russia where they bowed out with a quarter-final defeat by Belgium.
The goalkeeping position has been problematic for Liverpool under Klopp, with Loris Karius and Simon Mignolet struggling to hold down their first-team spots last season.
Karius endured a dismal night in last season's Champions League final, making two mistakes that led to goals in a 3-1 defeat to Real Madrid.
