Liverpool signed Brazil goalkeeper Alisson from Roma in a world record deal valued at 72.5 million euros by the Italian club.
"The 25-year-old signed a long-term contract with the Reds after undergoing a medical and completing the formalities of his switch at (training centre) Melwood today," said the Liverpool website.
Roma confirmed the value of the transfer for Alisson who featured at the World Cup in Russia.
"ASRoma can confirm that Alisson Becker has completed his move to Liverpool, in a deal that could be worth up to 72.5 million euros," the Serie A club said on Twitter yesterday.
Alisson is a key signing for Jurgen Klopp who was desperate for a new goalkeeper to replace Loris Karius following his calamitous Champions League final performance against Real Madrid.
Chelsea were also linked with Alisson who began his career with Internacional in Brazil before moving to Roma two years ago.
"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," Alisson told liverpoolfc.com.
"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."
"Yesterday, he sent me a text message saying: 'Hey, what are you waiting for?' As the negotiations were at an advanced stage, I replied to him straight away saying: 'Calm down, I'm on my way!'
Klopp added: "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."