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".
