By Daniel Sandford
BBC News, Washington
Mr Kennedy was worried that his death would derail healthcare reform
With Senator Edward Kennedy's funeral still a day away, discussions over his succession have already begun, in part prompted by the senator himself.
Under Massachusetts law, there should be a special election to fill his seat in the Senate between 145 days and 160 days after it becomes vacant.
That would mean a vote in January or February of next year.
But, in a letter he wrote just days before his death, Senator Kennedy said that the rules in Massachusetts should be changed because, in his words: "It is vital for this Commonwealth to have two voices speaking for the needs of its citizens and two votes in the Senate during the approximately five months between a vacancy and an election."
Everybody knows what his main concern was. Without his vote in the Senate, the Democrats' chances of getting their contentious healthcare reform through this autumn are badly diminished.
Universal healthcare was a long-cherished dream for Senator Kennedy, and the thought that his death might scupper the plans was clearly weighing heavily on his mind in his final days.
His seat is vital because of the nature of Senate arithmetic.
Under the "cloture" rule, if three-fifths of Senators support a motion, opponents are barred from blocking it by talking it out (a tactic known as a "filibuster"). Three-fifths of the 100-seat house is 60.
There are currently 40 Republican Senators, which means the Democrats need every one of their seats, as well as the Independents (Democrat-leaning Bernie Sanders and Joe Lieberman), to make sure any healthcare legislation is passed.
Of course there is no guarantee that all senators will vote along party lines but the Democrats are certainly going to need all the votes they can get.
Mr Kennedy's widow Vicki is reportedly not interested in taking his seat
In his deathbed letter, Senator Kennedy proposed that the Massachusetts governor should appoint an interim senator until the special election is held.
The current Governor, Deval Patrick, is a Democrat, and would presumably appoint someone sympathetic to the party's health care cause.
Governor Patrick has said he supports the idea of a rule change but it would need to be passed by the Massachusetts legislature, which is by no means guaranteed.
The next question is who might fill the seat. Will it be a Kennedy, continuing the political dynasty?
Speculation on family successors has centred on his widow Vicki (though the family says she is not interested) and his nephew Joseph Kennedy.
Joseph is the son of Robert Kennedy. A former member of the House of Representatives, he is currently running a charity, distributing discounted heating oil to the poor.
Beyond the Kennedy family, there is no shortage of potential candidates for the seat. Possible appointees include Michael Dukakis, who ran for the presidency in 1988, and former Labour Secretary Robert Reich.
Massachusetts is accustomed to having high-profile senators.
The other seat is held by John Kerry, who was the Democratic nominee for president in 2004.
Edward Kennedy's predecessor (though there was a brief an interim appointment) was his brother John, later President Kennedy.