Chelsea and Real Madrid had been linked with a move for the 25-year-old but the Liverpool have beaten both clubs to solve a problem area for Klopp.
Alisson told Liverpoolfc.com: “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.
“In terms of my life and my career, it’s a huge step for me being part of this club and this family. You can be certain that I’ll give my all.”
Alisson arrives at Anfield with a growing reputation as one of Europe’s best after an impressive 2017/18 season for Roma.
Klopp told Liverpoolfc.com: “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.
“It shows the value of goalkeepers, of course, in this moment. It will happen a lot in the next few weeks I guess and that’s it, so we are really happy to have him here now.
“His English is surprisingly good and he is a real personality. He has meanwhile a lot of experience in the last few years, in Europe and in Rome, he’s played there on an outstandingly high level and he did the same at the World Cup.
“It was for Brazil not too lucky, that’s how it is, but then he played a really good World Cup. So it’s just a good moment.”
He kept 17 clean sheets in 37 Serie A games for Roma and was Brazil’s No 1 at the recent World Cup.
Alisson’s transfer surpasses the £34.7m fee Manchester City paid Benfica for Ederson last summer.
Alisson has been Liverpool’s first choice to become their No 1 goalkeeper after manager Jurgen Klopp lost faith in Loris Karius.
Karius made two mistakes in the Champions League final defeat to Real Madrid but was backed by Klopp after the German was found to have suffered a concussion.
But an error in a pre-season friendly against Bury appears to have cost Karius and forced Klopp into the transfer market.
Alisson cut short a family holiday to complete his move to Liverpool and he left Merseyside this evening to rejoin them.
The goalkeeper will not take part in Liverpool’s pre-season tour of the US and will instead join up with his new team-mates at their four-day camp in France on July 30.
Roma bid farewell to Alisson in a tweet saying: “Thank you Alisson and good luck in your new adventure.”
Sporting director Monchi had said Liverpool’s offer was too good to turn down.
He said: “When a very substantial offer comes in, you have to consider it. We weighed up the pros and cons and decided to speak to Liverpool.”