The Bloody Sunday killings were unjustified and unjustifiable, the Prime Minister has said.
Thirteen marchers were shot dead on 30 January 1972 in Londonderry when British paratroopers opened fire on crowds at a civil rights demonstration.
Fifteen others were wounded. The Saville Report is heavily critical of the Army and found that soldiers fired the first shot.
Prime Minister David Cameron said he was "deeply sorry".
He said that the findings of the Saville Report were "shocking".
A huge cheer erupted in Guildhall Square in Derry as Mr Cameron delivered the findings which unequivocally blamed the Army for one of the most controversial days in Northern Ireland's history.
BBC legal affairs correspondent Clive Coleman said the decision whether or not to prosecute the soldiers would not be straightforward.
There needed to be sufficient evidence to provide a reasonable prospect of conviction - not an easy test after 38 years.
"If any defendent believes that the passage of time makes a fair trial impossible, they could argue the prosecution was an abuse of process," our correspondent said.
"Any prosecutions would also need to be judged to be in the public interest."
Speaking in the House of Commons, Mr Cameron said what happened on Bloody Sunday was wrong.
Army fired first shot
The Prime Minister said:
- No warning had been given to any civilians before the soldiers opened fire
- None of the soldiers fired in response to attacks by petrol bombers or stone throwers
- Some of those killed or injured were clearly fleeing or going to help those injured or dying
- None of the casualties was posing a threat or doing anything that would justify their shooting
- Many of the soldiers lied about their actions
- The events of Bloody Sunday were not premeditated
- Northern Ireland's Deputy First Minister Martin McGuinness, Sinn Fein, was present at the time of the violence and "probably armed with a sub-machine gun" but did not engage in "any activity that provided any of the soldiers with any justification for opening fire"
Mr McGuinness denied having a sub-machine gun. When asked about the Saville finding that it was probable that he had the weapon, he replied: "No". He said the report had cleared everybody in the city.
The head of the Army, General Sir David Richards, said he fully supported Mr Cameron's apology.
"The report leaves me in no doubt that serious mistakes and failings by officers and soldiers on that terrible day led to the deaths of 13 civilians who did nothing that could have justified their shooting," he said.
General Sir Mike Jackson, who served in the Parachute Regiment on Bloody Sunday said: "The Prime Minister made a fulsome apology and I join him in so doing."
But he said the Army's service in the 20 years after Bloody Sunday should be recognised.
"Northern Ireland is a very different place from what it was 40 years ago, not least because of this sacrifice and I ask that Lord Saville's report is seen in that context."
Stephen Pollard, a solicitor representing the soldiers said Lord Saville did not have the justification for his findings and accused him of "cherry-picking the evidence".
"There is just as much evidence for the opposite conclusion," he said.
Following the report, the decision to prosecute any individual soldier rests with Northern Ireland's Public Prosecution Service (PPS).
In a statement, the PPS said their director and Chief Constable Matt Baggott would consider the report to determine the nature and extent of any police investigations.
Referring to the agreement that witnesses to the inquiry could not incriminate themselves, the statement continued: "The undertaking given by the Attorney General in 1999 to witnesses who provided evidence to the inquiry will also require to be considered.
"It is not practical, at this stage, to say when such decisions will be taken other than to indicate that the matter will be considered as expeditiously as possible."
The report was commissioned in 1998 by the then Prime Minister Tony Blair under the auspices of former High Court judge, Lord Saville of Newdigate.
The Saville Inquiry took witness statements from hundreds of people and has become the longest-running and most expensive in British history.
It closed in 2004 with the report initially due for publication the following year.
It cost £195m and took 12 years to complete.
Thousands of people gathered outside the Guildhall to watch Prime Minister David Cameron deliver the report to Parliament on a huge screen in what was an emotional day for the city.
Earlier, crowds retraced the steps of the original marchers from the Bloody Sunday memorial in the city's Bogside close to the spot where many of the victims died.
According to BBC NI political editor Mark Devenport, while it may not have been the bloodiest day in the history of the Troubles, "the significance of that day in shaping the course of the conflict cannot be overstated".
"The actions of the Parachute Regiment in shooting dead 13 unarmed civil rights protesters immeasurably strengthened Irish republicans' arguments within their own community and provided the Provisional IRA with a flood of fresh recruits for its long war," he said.
Our correspondent also said Bloody Sunday set in train the suspension of the Northern Ireland government in March 1972, which led to the decades of direct rule from London.
The full process of restoring devolution was only completed in 2010.
An inquiry chaired by Lord Widgery was held in the immediate aftermath of the killings but it failed to satisfy families of the victims.
Clarification 9 April 2019: This article was amended to remove a reference to the death of John Johnston. This reflects the Bloody Sunday Inquiry's finding about Mr Johnston's death several months after he was wounded in Derry on 30 January 1972. The inquiry report states that his death was "not the result of any of the wounds he sustained on Bloody Sunday".