Dozens of former sub-postmasters and postmistresses should get a clear path in quashing their convictions for fraud, theft and false accounting.
They were convicted of stealing money after the Post Office installed a new computer system, with some imprisoned.
Many have now been told the Post Office will not contest their appeals against conviction.
For years, their criminal record put some of those affected in dire financial circumstances.
Some struggled to secure work, lost their homes, and even failed to get insurance owing to their convictions, but they always said the fault was in the computer system.
The Post Office said it would not oppose 44 out of 47 appeals. The decision makes it almost certain that convictions will be quashed, although Court of Appeal judges must decide and the Post Office can seek a retrial. More cases remain under scrutiny.
Jo Hamilton is one of the people who has been told that the appeal against her conviction will not be contested.
"Twelve years I've been battling and it's just drained us. We've been fighting to prove we were innocent. There it is in black and white. I'm not a criminal," she said.
"It's just gone on for so long. It's been a massive part of my life."
Mrs Hamilton was accused by the Post Office of taking £36,000 from the village shop she ran in Hampshire.
She said issues in the Horizon computer system led to big discrepancies in her accounts, which she reported to her Post Office area manager. But that manager could find nothing wrong with the system, and she was put in a situation where "you had to prove your innocence".
After a distressing two-year process, she eventually pleaded guilty to false accounting at Winchester Crown Court in order to escape a more serious charge of theft.
She had to give up her shop and found it difficult to get a new job due to her criminal record. Declaring the criminal record also affected everyday life, such as failing to get car insurance.
She made ends meet by doing cleaning jobs for people in her village who did not believe she was guilty. Many of them, including the local vicar, came to support her at her sentencing hearing.
Another former sub-postmistress, Seema Misra, was pregnant with her second child when she was convicted of theft and sent to jail in 2010.
She was in tears after being told her appeal would not be contested, saying that she had been "suffering" for 15 years as a result of the case.
"Prison was my worst nightmare. I never thought of giving up," she said.
"I'm so, so happy. Justice has been done. I can now proudly say my name."
The Post Office agreed in December to pay almost £58m to settle the long-running dispute with more than 500 sub-postmasters and postmistresses affected by the scandal, many of whom had been fired or made bankrupt as a result.
The settlement brought an end to a mammoth series of court cases over the Horizon IT system used to manage local post office finances since 1999.
Earlier this week, the government announced that retired judge Sir Wyn Williams would lead the inquiry into the Post Office's failings after repeated calls for a judge-led, full public inquiry.
What is the Horizon computer scandal?
The Horizon system, developed by the Japanese company Fujitsu, was first rolled out in 1999 to some post offices to be used for a variety of tasks including accounting and stocktaking.
But from an early stage it appeared to have significant bugs which could cause the system to misreport, sometimes involving substantial sums of money.
It was difficult for sub-postmasters to challenge errors because they were unable to access information about the software to do so.
After more than 900 prosecutions, 550 sub-postmasters raised civil actions against the Post Office which agreed to pay £58m damages.
The English Criminal Cases Review Commission (CCRC) has referred 47 convictions brought under Horizon evidence to the Appeal Court.
The Director of Public Prosecutions is also considering whether there should be charges of perjury against officials who claimed in court there were no problems with Horizon, even though emails and other documents now suggest they knew there were.
Solicitor Neil Hudgell, who represented 33 of the sub-postmasters, said the Post Office's decision not to contest appeals would have wider implications.
"[This] is a landmark moment, not only for these individuals, but in time, potentially hundreds of others. The door to justice has been opened," he said.
"We are today obviously delighted for the people we represent. Clearing their names has been their driving goal from day one, as their reputations and livelihoods were so unfairly destroyed."
Six cases which were only heard by magistrates do not go to the Court of Appeal but instead to another hearing at the Crown Court, but the Post Office will not offer any evidence, which will mean they will be cleared.
Tim Parker, chairman of the Post Office, said: "I am sincerely sorry on behalf of the Post Office for historical failings which seriously affected some postmasters.
"Post Office is resetting its relationship with postmasters with reforms that prevent such past events ever happening again."