43-year-old Italian Riccardo Tisci, who was creative director at the famed French fashion house Givenchy from 2005 to 2017, will take over the reins at Burberry on March 12.
He is credited with reviving Givenchy by cultivating links with celebrities, including by designing stage outfits for the tours of Madonna, Beyonce and Rihanna, and directing the artwork for an album by Jay Z and Kanye West.
Tisci will be based in London and direct all Burberry's collections, presenting his first in September, the company said.
Chief executive Marco Gobbetti said he was "delighted" at the appointment, adding: "Riccardo is one of the most talented designers of our time.
"His designs have an elegance that is contemporary and his skill in blending streetwear with high fashion is highly relevant to today's luxury consumer.
"Riccardo's creative vision will reinforce the ambitions we have for Burberry and position the brand firmly in luxury."
Tisci said he was honoured to take over and pleased to be reuniting with Gobbetti, who was chief executive of Givenchy between 2004 and 2008.
"I have an enormous respect for Burberry's British heritage and global appeal and I am excited about the potential of this exceptional brand," he said.
Bailey presented his final show at last month's London Fashion Week, bowing out after 17 years.