Photo: PA
Lawyers for the 57-year-old, who is dying of prostate cancer, said he had taken the unexpected decision after his condition took a "significant turn for the worse".
He is expected to be freed next week on compassionate grounds, but relatives of the victims had expected his appeal to continue, even after his death.
They reacted to the latest news with dismay and immediately renewed their calls for a full public inquiry into the atrocity in which 270 people died.
The decision means that Megrahi, who has always protested his innocence, will die a guilty man.
He had previously claimed that he would not return to Libya until he cleared his name, and politicians expressed concern that he may have been put under pressure as part of a deal to return home.
Scottish ministers are thought to be preparing to free him after being persuaded that he does not have long to live.
He lost his original appeal in 2002 but was granted a second appeal after an independent review body ruled that he may be the victim of a miscarriage of justice.
The Scottish Executive, which insists no final decision has been taken on the Libyan's fate, said it was not involved in the appeal decision.
Reports that Kenny MacAskill, the Scottish justice minister, has already taken the decision to free him next week have infuriated American relatives of the Lockerbie victims.
They are convinced of his guilt and believe he should die in jail in Scotland, where he has served only eight years of a minimum sentence of 27 years. Most of the British families believe he is innocent and support his release.
Lawyers confirmed that he had asked for the appeal to be dropped and the request will be heard in the High Court in Edinburgh on Tuesday.
Pamela Dix, whose 35-year-old brother Peter died in the bombing, said she was disappointed by the decision.
Speaking from Woking in Surrey, she added: "At the moment there is no other process or procedure ongoing to tell us how the bombing was carried out, why it was done, the motivation for it and who ordered it."
Martin Cadman, 84, of Burnham Market, Norfolk, who lost his son Bill, 32, said: "It's been nearly 21 years since the event and where are we? Nowhere."
Meanwhile, Daniel Kawczynski, the Tory chairman of the all-party parliamentary group on Libya, said Megrahi should not be freed until the killer of Pc Yvonne Fletcher Â who was shot dead outside the Libyan embassy 25 years ago - was handed over.
Memorial of , at Dryfesdale Cemetery 
, the only individual convicted in connection with the bombing of  in 1988, has been released by Scottish  , on compassionate grounds.

Megrahi is suffering from   and will be allowed to return to his home country of Libya.

270 people were killed when, on December 21, 1988, the  flight from London's Heathrow Airport to New York's  was destroyed by a bomb whilst in flight over southern Scotland.

A police convoy escorted Megrahi from his former prison home in  to , where he boarded an  flight to . He was told he could not remain in Scotland on security grounds.

In announcing the release on compassionate grounds, Justice Secretary MacAskill stated, "Al-Megrahi now faces a sentence imposed by a higher power. It is one that no court in any jurisdiction could revoke or overrule. It is terminal, final and irrevocable. He is going to die".

The conviction remains controversial. Last year then-president of US group ''Victims of Pan Am Flight 103'' told ''Wikinews'' that the vast majority were satisfied in Megrahi's guilt. ''UK Families Flight 103'' painted a very different picture to ''Wikinews'' of the opinions in Britain: "UK Families have different views about Megrahi's guilt or innocence.  Certainly some, including my husband and I, believe that we are not in a position to make a judgment about whether he was involved in some way or not," said group coordinator Jean Berkley, whose son was killed. "Much of the evidence at the trial was circumstantial and confusing and it is a fact that the Scottish Criminal Cases Review Commission, after considering the matter for three years, came to the conclusion that there were grounds for appeal. The UK families are united in believing that the full independent inquiry for which we have been asking since 1989 should now take place, to deal with the many unanswered questions and enable the evidence which would have emerged from the now abandoned appeal to be made public." This is in sharp contrast to '''', which earlier reported that the majority of British families felt Megrahi was innocent.

The United States government had been strongly opposed to any possible release. US Secretary of State Hillary Clinton had previously called the possibility "absolutely wrong". MacAskill sought to justify the decision by commenting that "Compassion and mercy are about upholding the beliefs that we seek to live by, remaining true to our values as a people - no matter the severity of the provocation or the atrocity perpetrated."

Megrahi issued a statement shortly before leaving HMP Greenock in which he maintained his innocence. "The remaining days of my life are being lived under the shadow of the wrongness of my conviction. I have been faced with an appalling choice: to risk dying in prison in the hope that my name is cleared posthumously or to return home still carrying the weight of the guilty verdict, which will never now be lifted. The choice which I made is a matter of sorrow, disappointment and anger, which I fear I will never overcome," said Megrahi.

Robert Gibbs, press secretary for the White House, said "The United States deeply regrets the decision by the Scottish Executive to release Abdel Basset Mohamed al-Megrahi. As we have expressed repeatedly to officials of the government of the United Kingdom and to Scottish authorities, we continue to believe that Megrahi should serve out his sentence in Scotland." Many US victims have also reacted with anger. 

Many US victims' families have reacted with anger. One relative commented that "This might sound crude or blunt, but I want him returned from Scotland the same way my wife Lorraine was and that would be in a box." Dr. , a UK victim, disagreed, saying Megrahi had "nothing to do with" the disaster and calling the earlier dropping of the appeal "a blow to those of us who seek the truth."

Megrahi's plane was greeted by crowds in Tripoli waving both Libyan and Scottish flags. , son of Libyan leader , held his hand as he exited the aircraft amid heavy security. Loudspeakers broadcast patriotic music and it is reported that celebrations are ongoing in Tripoli.