Deloitte, which is handling the administration, said the collapse was driven by competition from internet firms and digital streaming of movies and games.
The Uxbridge-based business opened its first store in London in 1989 and now trades from 528 outlets employing 4,190 staff.
It has more than 300 workers in Scotland, working in 66 shops.
Blockbuster will continue to accept gift cards and credit bought through its trade-in scheme for second-hand movies and games, as well as operating its loyalty scheme.
The collapse comes a day after another music and entertainment retailer, HMV, hit the rocks, with more than 4,000 jobs under threat.
Deloitte is handling both administrations.
More than 8,000 retail jobs have been put at risk during the last two days alone in a bleak week for Britain's high street.
Lee Manning, joint administrator and partner in Deloitte's restructuring services practice, said: "In recent years Blockbuster has faced increased competition from internet-based providers along with the shift to digital streaming of movies and games.
"We are working closely with suppliers and employees to ensure the business has the best possible platform to secure a sale, preserve jobs and generate as much value as possible for all creditors."
Deloitte stressed Blockbuster's core business was profitable and they would be looking for a rescue deal for all or part of the business as a going concern.
Blockbuster's US parent went bankrupt in 2011, but was rescued by US pay-TV provider Dish Network in a 320 million US dollar (£200 million) deal, which saved hundreds of stores from closing and prevented tens of thousands of US job losses.
The UK group has sought to branch out in recent years in an attempt to revive business, including by expanding its online rental offering while also launching a trade-in service for pre-owned titles, allowing members to swap second-hand games and movies for cash or credit on their membership accounts.