An initial official government probe conducted within weeks of the shootings concluded that the soldiers were blameless.
The shootings helped galvanise support for the Provisional IRA.
The Ministry of Defence has confirmed it will cover the legal costs of any soldier facing criminal charges over Bloody Sunday.
"The decision to prosecute just one ex-soldier does not change the fact that Bloody Sunday was a massacre of innocents", Sinn Fein's Northern Ireland leader Michelle O'Neill said in a statement.
British Defence Secretary Gavin Williamson said the UK Government would fund all of the former soldier's legal bills. This includes funding all his legal costs and providing welfare support.
"Our serving and former personnel can not live in constant fear of prosecution", he said.
One former British soldier to face charges over Bloody Sunday killings
But 16 other ex-paratroopers and and two former members of the Official IRA will face no action. "In these circumstances the evidence Test for Prosecution is not met".
Solicitor Ciaran Shiels, a solicitor for a number of them, said they were "disappointed that not all of those responsible are to face trial", PA reported.
"The innocence was gone; also the possibility of a reformist. solution to what was going on in Northern Ireland was, if not destroyed, substantially diminished by Bloody Sunday", McCann said.
Nonetheless, he said the relatives were happy for the families of the six victims who will now see a soldier prosecuted. The victims were all unarmed Catholics.
As well as the 13 who died on the day, 15 others were shot and injured.
The soldiers involved were members of the 1st Battalion, Parachute Regiment, also known as "1 Para".
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Prosecutors had been considering evidence in relation to counts of murder, attempted murder and causing grievous injury with intent.
The inquiry was authorised by then-Prime Minister Tony Blair in 1998 ahead of the negotiations that led to the Good Friday peace accord.
The charges follow a decade-long investigation that concluded the soldiers killed unarmed demonstrators.
Following the report, UK Prime Minister David Cameron issued a full State apology on behalf of the UK for "unjustified and unjustifiable" events.
The charges announced yesterday come more than two years after police referred their findings to prosecutors and nearly nine years after the conclusion of the Bloody Sunday Inquiry, which was tasked with determining what happened, not bringing criminal charges. One has since died.
"All four soldiers insisted they had shot at people carrying bombs or firearms - claims rejected by Saville".
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More than 660 witness statements were taken and investigators gathered photos as well as video and audio evidence.
"The Bloody Sunday families are not finished yet", he said.
But the PPS decisions in respect of potential perjury charges will be announced at a later date.
Standing in the great throng outside Londonderry's Guildhall that fine June morning in 2010 when the Saville report was published, it was impossible not to be moved by the sheer joy of those who had campaigned for 38 years for the truth about Bloody Sunday.
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