Visa is operating at "full capacity", the company says, hours after millions of customers were left unable to pay for goods and services.
The firm apologised as it ruled out "malicious" activity as the cause of the disruption.
There were reports of "chaos" at supermarket tills as customers abandoned their shopping when they were unable to pay for it.
A statement on the Visa Europe website, posted in the early hours of Saturday, said: "Visa Europe's payment system is now operating at full capacity, and Visa account holders can now use Visa for any of their purchases and at ATMs, as they normally would."
For around six hours on Friday, businesses across Europe were unable to process some payments made using Visa cards, impacting potentially tens of millions of customers.
My card was declined on contactless payment at the chippy (fish supper on #nationalfishandchipday of course!) so I put it through on chip and pin. Checked my online banking and the “declined” payment was charged so people need to check they aren’t being double charged #visa— Ian Little (@ianjlittle) June 1, 2018
Visa admitted it had not achieved its goals of having its network operating all of the time and said the fault was "the result of a hardware failure".
£1 in every £3 spent in the UK is on a Visa card.
The extent of the problem, which Visa called a "service disruption", has not yet been clarified. Visa has also not commented on whether retailers will be compensated for lost earnings.
Some customers have reported that funds appear to have been taken from their accounts, despite payments being declined in-store.
Retailers including Marks & Spencer and Sainsbury's confirmed they were affected, but others, including Transport For London did not report any problems from Visa contactless users on their services.
Some retailers resorted to putting up notices warning customers that they were unable to take card payments.
Business owners expressed anger that they had lost sales.
Sandra Foy, who owns a bookshop in Manchester, told Sky News: "I run a small business and the loss of any business is a big deal for us. Not knowing when this is going to be sorted out is incredibly frustrating."
Worldpay, the UK's leading payments provider, said: "Customers may be seeing intermittent transaction declines due to an issue Visa is currently experiencing in Europe with all acquirers.
"We will provide an update as soon as we have one."
Lloyds Bank said: "We are aware of an industry wide issue effecting Visa payments which is under investigation.
"ATM and Mastercard transactions are not impacted. We are working to resolve the issue as quickly as possible."
Customers used Twitter to vent about being unable to pay or being forced to get cash out.
Manchester-based user MancBee posted a picture of a sign on the front of an Aldi supermarket that informed customers card payments were down.
Another Twitter user wrote: "We've just been shopping and my Visa card didn't work. Thankfully it did work at the cash machine, on the side of the bank. We had to pay in cash. Bloody nuisance."
Dan Heley, 30, from Luton and living in Weston-Super-Mare, posted a photo of KFC warning customers that they were unable to accept cards.
He told Sky News: "In Morrison's there were frustrated parents with crying kids because the mum couldn't buy the tea for tonight, it's totally unacceptable. I wouldn't be surprised if Russian hackers were to blame."
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