David Cameron has said he is profoundly sorry for the "double injustice" of the Hillsborough football disaster.
Speaking after an independent report into previously unseen documents about the tragedy, the prime minister said police had failed to do enough and had also tried to blame Liverpool fans.
Ninety-six fans died after a crush at Sheffield Wednesday's ground in 1989.
Campaigner Trevor Hicks said the report showed a faster response from emergency services could have saved lives.
Mr Hicks, who lost two daughters in the disaster and is a member of the family support group, said it would now press for criminal action against those involved in the disaster.
"We feel a breakthrough has been made. The truth is out today and the justice starts tomorrow," he said.
The report has been compiled by the Hillsborough Independent Panel, which scrutinised more than 450,000 pages of documents over the last 18 months.
The victims' families have always challenged the original inquest, which concluded all the victims were dead or brain dead 15 minutes after the game had kicked off at 15:00.
Anne Williams claims her son Kevin was still alive at 16:00, and has called for the government to open a new inquest into his death.
By analysing post-mortem test results, the panel found 28 of the 96 victims had no "obstruction of blood circulation" and there was "separate evidence that, in 31, the heart and lungs had continued to function after the crush".
The medical advisor on the panel, Dr Bill Kirkup, said up to 41 of the 96 who died could have potentially been saved if they had received treatment earlier.
"In total, 41 people therefore had potential to survive after the period of 3:15. What I can't say is how many of those could have been saved," he said.
Relatives of the Liverpool supporters who died at Hillsborough were handed the report at Liverpool Anglican Cathedral, ahead of the media.
Mr Hicks said the families gave the panel a standing ovation when it finished reporting its findings to them and three people fainted as the information came out.
Margaret Aspinall, chairwoman of the Hillsborough Families Support Group, said what the families had gone through was an "absolute disgrace".
"They were the liars and we were the truthful ones," she said.
"It doesn't make us feel better, because we will always be the losers at Hillsborough."
The Hillsborough Justice Campaign has also welcomed the prime minister's apology.
Mr Hicks said: "We are staggered at the level of incompetence. We don't blame the guys on the shop floor, the common bobby, the problems were the command structure."
The report comes after 23 years of campaigning from Liverpool fans and relatives of the victims to find out exactly what happened on the day of the disaster, which saw the biggest loss of life at any UK sporting event.
Mr Cameron told the House of Commons the panel found the safety of the crowds at Hillsborough had been "compromised at every level".
He said "deficiencies" at the ground were well known and it failed to meet minimum safety standards.
The prime minister apologised for the double injustice, which was both in the "failure of the state to protect their loved ones and the indefensible wait to get to the truth", and in the efforts to denigrate the deceased and suggest that they were "somehow at fault for their own deaths".
The report showed police and emergency services had made "strenuous attempts" to deflect the blame for the disaster on to fans.
The panel said it found evidence the police's submissions to the original inquiry led by Lord Chief Justice Taylor "emphasised exceptional, aggressive and unanticipated crowd behaviour".
It said the first inquiry also emphasised "large numbers of ticketless, drunk and obstinate fans involved in concerted action, even 'conspiracy', to enter the stadium".
The report also found South Yorkshire Police had changed some of the 164 statements made in the wake of the tragedy.
It found 116 of the police statements identified for "substantive amendment" had been "amended to remove or alter comments unfavourable to South Yorkshire Police".
South Yorkshire Police Chief Constable David Crompton said on Wednesday he wanted to offer his "profound apologies".
He said: "I'm absolutely shocked.
"If you put yourself in the position of the Hillsborough families, 96 didn't come home and in the immediate aftermath, when police lost control, lies were told about how that happened and then, later in the day, you had to identify your loved one in a makeshift mortuary.
"That adds up to about the worst possible set of circumstances anyone could imagine."
Mr Cameron said there were three main areas highlighted in the report - failures by the authorities in protecting those at the ground, an attempt to blame the fans and doubt cast on the original coroner's inquest.
The prime minister said the independent panel's review found:
- New evidence about how the authorities failed, including documents which show a delay from the emergency services when people were being crushed
- Shortcomings in the response by the ambulance service and other emergency services in addition to failings by police
- Rescue attempts were held back by failures of leadership and co-ordination
- Victims' families were correct in their belief that some of the authorities attempted to create a "completely unjust" account of events that sought to blame the fans
- "Despicable untruths" about the behaviour of fans were part of police efforts "to develop and publicise a version of events that focused on allegations of drunkenness, ticketlessness and violence"
- Police officers carried out police national computer checks on those who had died in an attempt "to impugn the reputations of the deceased"
- No evidence of any government trying to conceal the truth
Mr Cameron said Attorney General Dominic Grieve would review the report as quickly as possible in order to decide whether to apply to the High Court to order a new inquest.
Kelvin MacKenzie, editor of the Sun newspaper when it ran a story blaming fans, offered his "profuse apologies".
He wrote the headline The Truth on the controversial front page report, published in the days following the disaster, which alleged fans had picked pockets of victims, urinated on police and beat up officers trying to save lives.
In a statement he said: "I published in good faith and I am sorry that it was so wrong."
But Mr Hicks rejected his apology as "too little, too late".
The Sun's current editor Dominic Mohan also added his apologies, saying: "Twenty-three years ago The Sun newspaper made a terrible mistake.
"We published an inaccurate and offensive story about the events at Hillsborough. We said it was the truth - it wasn't."
Liverpool FC chairman Tom Werner said: "The world has heard the real truth about what happened at Hillsborough."
Former Liverpool manager Kenny Dalglish also welcomed the publication of the documents.
Mr Dalglish, who was in charge of the side on 15 April, 1989, said on Twitter: "Very positive outcome. 23 yrs waiting for the truth next step justice."
Liverpool Walton Labour MP Steve Rotheram said it was a "momentous day for Liverpool".
He said: "Finally, we have the undeniable truth. The truth that many innocent people could and should have been saved.
"A truth that unequivocally confirms that Liverpool fans were not the cause of the disaster and that drink was not a significant factor."
Sheffield Wednesday has also issued an apology to all the families whose relatives were involved.
A number of the victims' families have been campaigning for the Hillsborough documents to be released for more than 20 years.
Cabinet papers are not usually published in the UK until 30 years after they have been written but MPs agreed to their full, uncensored disclosure last year.
'Maximum possible disclosure'
Approval came after 140,000 people signed an e-petition, set up by Liverpool fan Brian Irvine, to trigger a House of Commons debate on the issue.
The panel, chaired by the Bishop of Liverpool the Right Reverend James Jones, has now analysed documents relating to the disaster from more than 80 organisations.
Bishop Jones said: "Our job has simply been to oversee the maximum possible disclosure of all the documents and to write a report which adds to public understanding and therefore our terms of reference don't actually allow us to make any recommendation.
"The documents speak for themselves."
Ninety-five fans were crushed to death and hundreds more injured on the overcrowded terraces of the Hillsborough stadium, which was hosting an FA Cup semi-final between Liverpool and Nottingham Forest.
The 96th victim, Tony Bland, was left in a coma after the disaster and died in 1993.
An independent inquiry led by Lord Chief Justice Taylor found the main cause of the disaster was a failure in crowd control by South Yorkshire Police.
But the victims' families wanted to know to exactly what caused the tragedy and what happened in the aftermath.
People in Liverpool were asked to observe a two-minute silence as a mark of respect to those who died.
During the silence - held at 15:06 BST to mark the time the game was stopped - the bells at Liverpool Town Hall Municipal Buildings on Dale Street and Liverpool Parish Church rang out 96 times.
A vigil to mark the release of the papers at St George's Plateau near to Liverpool Lime Street station began at 18:00 BST.
Victims' relatives and campaigners have been making speeches, as well as representatives of public bodies including Shadow Secretary of State for Health Andy Burnham MP and Liverpool Mayor Joe Anderson, lanterns were lit and songs, including You'll Never Walk Alone, have been sung.