Cook, England's leading scorer in Tests with 11,057 runs, became captain in August 2012 and led his country to Ashes glory on home soil in 2013 and 2015 as well as series wins in India and South Africa.
Yorkshire batsman Joe Root is the early favourite to take over with an announcement expected in the next fortnight.
"It's been a huge honour to be England captain and to lead the Test team over the past five years," Cook, 32, said in a statement.
Cook, who also led England in a record 69 one-day internationals between 2010 and 2014, is the country's most capped Test skipper and has scored more Test centuries than any previous captain.
He was named Wisden Cricketer of the Year in 2012 and ICC World Test Captain in 2013.
He tendered his resignation to England and Wales Cricket Board (ECB) chairman Colin Graves on Sunday, but intends to continue playing Test cricket.
"It's a sad day personally in many ways, but I want to thank everyone I've captained, all the coaches and support staff and, of course, the England supporters and the Barmy Army who follow us home and away and have given us unwavering support," Cook added.
Cook's decision to stand down follows England's 4-0 series defeat in India at the end of last year.
The ECB said a "proper process" had started to appoint England's next Test captain.
England's director of cricket Andrew Strauss said Cook had made a "fantastic contribution" and "deserves to be seen as one of our country's great captains".
"We now move on with the process of appointing the right successor," Strauss said.
"We expect to be able to make an announcement before the team head to the West Indies on 22nd February.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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