The 23-year-old has signed a seven-year deal at Stamford Bridge, where he will take over between the sticks from Belgium's Thibaut Courtois.
Arrizabalaga has played 53 La Liga matches for Athletic Bilbao since breaking into the team in 2016.
Upon completing his move to Chelsea, he said: 'It's a very important decision for me, for my career, and also for my personal life.
He is the second key player to leave the club this year, after Manchester City signed French defender Aymeric Laporte for £57m in January.
The goalkeeper has moved for a fee in the region of £31m and has signed a six-year contract. Considered Spain's goalkeeper of the future, he was a reserve at the World Cup in Russian Federation.More news: NASA Introduces Nine Astronauts for First Commercial Flights
He joined Athletic at the age of nine.
Following loan spells with second-tier Ponferradina and Real Valladolid, he finally became a starter for Athletic early in the 2016-17 season, quickly establishing himself by showing impressive poise and maturity.
Having been a regular in the Spain Under-21s team over the last three years, as well as earning a cap for the national team in a friendly against Costa Rica in November 2017, Kepa said he had plenty of friends at Chelsea.
New Chelsea goalkeeper Kepa Arrizabalaga has revealed that he wants to win many titles and go down in Blues history.
Meanwhile, Kepa will replace the Belgium global in a deal that will see him become the world's most expensive goalkeeper, surpassing the £65m paid by Liverpool to Roma for Brazil goalkeeper Alisson Becker last month.
AS said Modric, who has not commented publicly, "foresaw joining Inter".