The banking industry has pulled out of the appeal to overturn the High Court ruling on the mis-selling of payment protection insurance (PPI).
The British Bankers' Association said it and the banks "do not intend to appeal" the judgment from April.
The Lloyds Banking Group was the first institution to back down last week, prompting other high street banks, headed by the BBA, to follow.
Shares in Lloyds fell 7.7% after it set aside £3.2 billion to repay claims by customers.
Barclays Bank says it will reserve £1 billion to pay PPI compensation and it will begin to process all on-hold complaints. Chief executive Bob Diamond apologised to customers and said: "We don't always get things right for our customers; when we get them wrong, we apologise and put them right."
Royal Bank of Scotland confirmed it would not be appealing the case but has yet to announce how much it will set aside for this.
HSBC has yet to make a statement on the issue.
Read: How to reclaim your PPI premiums
The news was welcomed by the Financial Ombudsman Service, Natalie Ceeney, chief ombudsman, says: "It's very good news that the banks will not be appealing the High Court's clear-cut judgment.
"We will be working with the banks, over the coming weeks, to ensure that consumers' complaints are dealt with fairly and promptly."
The banks will now begin to process complaints put on hold.
Stefan Maryniak, spokesperson at uSwitch.com, says by retreating from the battle, the banks have admitted they were wrong in this mis-selling scandal. They are now going to have to suffer the consequences of their actions, with compensation costs rising to the billions.
"There is still a huge amount of trust to be rebuilt between banks and consumers, so even though today's move is a step in the right direction, it could be too little too late. With all compensation cases now being looked at, we finally have a victory for consumers and common sense," he adds.
Were you mis-sold PPI?
If you answer yes to any of the following questions you could be able to make a claim:
Was it made clear to you that the PPI was optional?
If you feel you were forced to take out cover then you could have been mis-sold.
Was it explained to you that past medical conditions mean you are not fully covered?
Many policies do not cover existing medical problems so if you have to be off work due to an existing problem your PPI may not pay out.
Were you unemployed at the time you took out the PPI?
Most PPI policies include unemployment cover. If you were unemployed or retired at the time then this cover is of no use to you.
Were you self-employed when you took out PPI?
To have made a claim for unemployment if you were self-employed you would have had to declare yourself bankrupt or notify HMRC that you have stopped working.
Casual workers and those working on short term contracts also would not be eligible to claim. If this wasn't made clear to you, you could be entitled to claim.