The family of murdered police officer Yvonne Fletcher have said they are "deeply disappointed and frustrated" a suspect will not be prosecuted after key evidence was withheld for national security reasons.
The Libyan national, who is in his 50s, has been told the investigation against him will not proceed "at this time", the Metropolitan Police said.
Pc Fletcher was killed as she policed a demonstration against Colonel Muammar Gaddafi outside the Libyan People's Bureau in St James's Square on April 17 1984.
The shooting sparked a 10-day siege of the building before 30 of the occupants were deported back to Libya.
No-one has ever been prosecuted over the killing.
A family statement said: "We understand that some available evidence could not be used in court but are satisfied that the Metropolitan Police has left no stone unturned in its pursuit of justice in Yvonne's case.
"The family would like to thank the Met for its continued hard work and diligence and also for always keeping us informed at every turn.
"We are deeply disappointed and frustrated that a prosecution cannot proceed at this time.
"We had hoped that the latest turn of events would finally lead to some closure for the family.
"The family will make no further comment at this time."
The suspect was arrested in November 2015 by counter-terrorism officers in south-east England in what police described as a "significant turning point" in the inquiry.
But the force said that despite officers pursuing hundreds of lines of inquiry, including visits to Libya, some crucial evidence was inadmissible in court.
A spokesman for the Met said: "We believe our investigation has identified enough material to identify those responsible for WPc Fletcher's murder if it could be presented to a court.
"However, the key material has not been made available for use in court in evidential form for reasons of national security.
"Therefore, without this material and following a review of all the evidence that was available to prosecutors, the Crown Prosecution Service, who we worked closely with throughout, have informed us that there is insufficient admissible evidence to charge the man."
Assistant Commissioner Mark Rowley said: "I regret that we have not been able to deliver the justice that the victims and their families deserve.
"Our thoughts today are with WPc Fletcher's family and all those affected by the events of that day in 1984."
The Met statement added the investigation into the "act of state-sponsored terrorism" remained open but it was unlikely more evidence would be found.
"Although our investigation has always remained open, cases like this do become harder to solve over time," the force said.
"Our judgment is that this concludes what was by far the best opportunity to solve this tragic case and provide a degree of closure for the victims and their families.
"This investigation will never be closed but the likelihood of finding further evidence, in Libya or elsewhere, is low."
A CPS spokesman said: "After considering all the admissible and reliable evidence submitted by the MPS, in accordance with the Code for Crown Prosecutors, the CPS has concluded that there is insufficient evidence to provide a realistic prospect of conviction."
Home Secretary Amber Rudd praised the "hard work" of investigators.
She said: "I cannot comment on the details of this case.
"WPc Fletcher was one of their own.
"Her murder remains as shocking and senseless as the day it occurred and I understand that the decision will be deeply disappointing and frustrating for all her family, friends and colleagues."