Now, investigators say, there is renewed hope that the mystery may be solved. The authorities are awaiting results of DNA tests not available two decades ago and are completing a reinvestigation of the long-dormant case. ''We're expecting some sort of resolution,'' said Jonathan Benedict, the State's Attorney for Fairfield County.
If the killer is charged, it will largely be because of events set in motion by another notorious case: the William Kennedy Smith rape trial. The 1991 trial of Mr. Smith, a nephew of Senator Edward M. Kennedy, rekindled interest in Miss Moxley's unsolved murder because a nephew of Senator Robert F. Kennedy had been investigated as a possible suspect.
The investigation got another nudge recently when Mark Fuhrman, the former Los Angeles police detective, arrived in town to conduct his own investigation for a book he is writing about the murder. Mr. Fuhrman, who gained notoriety for his testimony during the O. J. Simpson trial and who later pleaded no contest to perjury charges, said his book would name the killer -- a person of power, wealth and influence. The Greenwich police have not cooperated with Mr. Fuhrman, and investigators discount his inquiry as irrelevant.
But after years with no movement on the case, Dorthy Moxley, Martha's mother, said help from just about any quarter was welcome.
''There was a period of 10 years when nothing happened,'' Mrs. Moxley said. ''The only way this is possibly going to be solved is to keep it before the public. There's always the hope it's going to reach some sort of closure.''
If nothing else, Mr. Fuhrman was instrumental in bringing Dr. Michael Baden, the former New York City Chief Medical Examiner, into the case. At Mr. Fuhrman's suggestion, Mrs. Moxley asked Dr. Baden, a family friend, to review the autopsy report on her daughter's body. Dr. Baden had testified as a defense witness in the Simpson case.
Dr. Baden said he had reviewed the documents and met with lawyers and investigators involved in the case but was waiting to see photographs of the body, which have disappeared from the file. Investigators are trying to track them down, he said.
Meanwhile, DNA testing may provide some clues, he said. ''There's a lot more that can be done with trace evidence now that couldn't be done 20 years ago,'' he said. ''It all depends on how carefully the trace evidence was collected and how carefully it was preserved.'' The authorities are waiting for the results of those tests.
Mrs. Moxley said she believed the authorities were testing her daughter's clothes for traces of saliva and other bodily fluids.
On one point, at least, there has been general agreement almost from the beginning. Miss Moxley, a popular and pretty teen-ager who loved tennis, swimming and her cats, died on the evening of Oct. 30, 1975, in a frenzy of blows from a golf club that had belonged to Ann Reynolds Skakel, Ethel Kennedy's sister-in-law. The blows had been so powerful that the club had shattered into three pieces, one of which was used to stab Miss Moxley though the neck.
Though her underpants had been pulled down, the authorities said there was no obvious evidence that she had been sexually attacked.
Mrs. Skakel, who had died two years earlier, had been the wife of Ethel Kennedy's brother, Rushton W. Skakel, who, with their seven children, lived near the Moxleys in Belle Haven, a gated community of estates that has a private security force. Thomas, the Skakels' 17-year-old son, was the last person known to have seen Miss Moxley alive.
On the night she died, Martha left home about 7 o'clock with some friends, her mother recalled. About two hours later, Martha and her friends stopped at the Skakel house and joined a group of teen-agers, but by 9:30 only Thomas Skakel and Martha remained.
Mrs. Moxley heard some commotion and barking dogs about 10 P.M., which was later determined to be about the time Martha was killed.
Thomas Skakel has said from the beginning that when he left Martha outside his house, she was alive, and that he does not know who killed her.
The Moxleys and the Skakels had been friends almost from the time the Moxleys moved to Greenwich from California a year earlier. David Moxley, Martha's father, had been promoted to head the New York office of Touche Ross, the accounting firm.
Initially, the Skakels cooperated with the Greenwich police, but several months after the murder, when the police requested records from Thomas Skakel's school, the Skakels withdrew their cooperation and have since declined to answer any questions.
Emanuel Margolis, the Skakels' lawyer, said he had no comment on recent activity in the case except to deny that Thomas Skakel had anything to do with Miss Moxley's murder. ''We're not going to be commenting on Mr. Fuhrman's book,'' he added. ''Everybody knows who he is.''
Mr. Fuhrman began a writing career with a book about the Simpson case, ''Murder in Brentwood.'' During the Simpson trial, he had become friendly with the author Dominick Dunne, who was covering the case for Vanity Fair. It was Mr. Dunne who suggested to Mr. Fuhrman that he write his next book about the Moxley murder. Mr. Dunne's novel ''A Season in Purgatory,'' based on the Moxley case, had been a best seller when it was published in 1993.
Mr. Fuhrman said he had made use of a confidential report by a private investigation concern that had been hired by the Skakels in an attempt to clear Thomas. That report was never made public, but Mr. Dunne had obtained a copy while working on his own book and gave the report to Mr. Fuhrman, he said. ''I have an embargo, but I can tell you that we will somewhat turn the case on its head,'' Mr. Fuhrman said. His book, ''Murder in Greenwich,'' is to be published in April.
Asked whether his involvement might hurt the case, Mr. Fuhrman said: ''How could it hurt? The police haven't done a thing in 22 years.''
Mrs. Moxley said there had been no contact between her family and the Skakels since shortly after the murder. ''These people have not cooperated in all these years,'' she said. ''I think the police held them in awe. They were looking at everybody else but them.''
Two years after Martha's murder, the Moxleys moved to Manhattan to get away from Greenwich. ''I know there are people out there who must know about this,'' she said. ''The people who know just won't come forward.''
In 1988, David Moxley died of a heart attack. ''He kept everything in,'' said Mrs. Moxley, who now lives in New Jersey, near her son and his family. ''He just couldn't talk about things like this.''