In his inaugural address, President Barack Obama signalled conciliation to America's foes by using the metaphor of an outstretched hand to an unclenched fist.
Already, there are signs that some were listening, sensing an opening for improved relations after eight combative years under President George W. Bush. Fidel Castro is said to like the new American leader, and North Korea and Iran both sounded open to new ideas to defuse nuclear-tinged tensions.
Unclear is what they will seek in return and whether they will agree to the compromises the U.S. is likely to insist on in exchange for warmer relations.
Are the Castro brothers willing to move toward democracy? Can Russia and the West heal their widening estrangement? Will Venezuelan President Hugo Chavez stop being a thorn in America's side?
Analysts say they see many bright opportunities amid the perils and complexities.
"In this dangerous world, in a world where America's leverage has either been exposed as not being as strong as the rhetoric implied or where it is simply diminishing, I think Obama's approach is exactly what America needs," said Robin Niblett, the director of Chatham House, a London-based think-tank.
"We need an approach that gets others to show their hand, that makes others have to think harder about their diplomacy, rather than just to react to rather strident and fixed American positions."
Nowhere are the opportunities -- and the risks -- of diplomacy more stark than in the Middle East.
Iran still considers the U.S. the "Great Satan," but a day after Obama was sworn in, it said it was "ready for new approaches by the United States."
Foreign Minister Manouchehr Mottaki said his country would study the idea of allowing the U.S. to open a diplomatic office in Tehran, the first since the 1979 Islamic Revolution.
Still, the differences between the two countries are deep -- U.S. suspicions about Iran's nuclear program, President Mahmoud Ahmadinejad's view that Israel should cease to exist, and Tehran's support for armed groups in Iraq, Lebanon and elsewhere. Analysts say that baby steps are all that can be hoped for, at least in the short term.
Iran's parliament speaker, Ali Larijani, said the country had doubts that Obama's Mideast policy will be different from the Bush administration, state television reported Sunday.
And diplomacy has its limits: Some observers believe feel the Islamic Republic is committed to developing nuclear weapons, whatever the cost. Tehran itself says its progam is aimed only at the peaceful production of nuclear power.
Israel and the Palestinians present the new president with one of his greatest challenges, and he has been quick to demonstrate his interest.
With the latest Gaza fighting still reverberating around the world, Obama appointed George Mitchell, mediator of peace in Northern Ireland, as special envoy to the Middle East.
While the task is daunting, some see room for Obama to manoeuvre.
Syria, often portrayed by Washington as a pariah state, has indicated it seeks no further quarrel with Washington, even saying it would like the new administration to mediate its stalled peace talks with Israel.
If Obama helps Syria, analysts say it could have far-reaching benefits. Syria has great influence over two of Israel's main enemies -- Hezbollah in neighbouring Lebanon, and Hamas, whose leaders live in Damascus. Assad could also help pressure Syrian ally Iran to take a more pragmatic stance.
Another striking sign of change on Obama's first full day in office was an article in the New York Times written by Libyan President Moammar Gadhafi.
In it, Gadhafi recognized the Jews' long history of persecution and said they "deserve their homeland."
His solution -- a binational Arab-Jewish state -- is a non-starter to Israelis. But his conciliatory language marked a striking turnaround from the firebrand Arab nationalist of old.
Elsewhere, some see hope for progress in the frustrating on-again-off-again talks with North Korea.
Hours before Obama's inauguration, a newspaper considered a mouthpiece for the North Korean government said Pyongyang would be willing to give up its nuclear arsenal if the U.S. accepts its conditions, which include establishing diplomatic relations.
Another crucial area for diplomacy is to repair the relationship with Moscow, riddled with distrust over Bush's plans to put missile interceptors in Europe and his support for giving NATO membership to Ukraine and Georgia.
Obama has been noncommittal about deploying a missile shield in Poland and the Czech Republic, and Democratic supporters of the new president have voiced a willingness to reopen talks on arms control.
But experts say it will be Washington's stance on NATO expansion that will determine future relations.
Sergei Karaganov, chairman of the presidium of Russia's Council on Foreign and Defence Policy, saw no room for Moscow to compromise because it feels it has already given away too much with nothing to show for it.
"The distrust, I believe, is almost complete after the years of the Bush administration and the previous years of Clinton," Karaganov said. "Compromise, constructiveness, any kind of good gestures toward the U.S. doesn't bring anything."
For its part, Washington has been rattled by the ferocity of Russia's war with Georgia, and by its hardline tactics in shutting off natural gas to Europe during a dispute over gas payments with Ukraine.
In Washington's own hemisphere, the greatest hope for diplomatic progress lies with Cuba, where Fidel Castro and his brother-successor, Raul, have both spoken positively about Obama.
On Wednesday, Argentina's president came out of a meeting with Fidel Castro quoting him as having said Obama seems "like a man who is absolutely sincere."
That's quite a change from Cuba's attitude toward Bush, who was depicted on Havana billboards as a vampire.
Analysts say Obama is unlikely to move quickly to end Washington's embargo on the island, which he has said will remain in force as a bargaining chip.
Wayne Smith, a former head of the U.S. Interests Section in Havana, recommends a step-by-step approach, first lifting restrictions that keep most Americans from visiting the island, before insisting on greater democracy and the freeing of political prisoners.
"Cubans are not going to accept these conditions. They're not going to free political prisoners on a promise that you're going to do something," he said. "Free the travel restrictions and start negotiations."
Some enmities seem intact despite the change of administrations.
Venezuelan President Hugo Chavez has stepped up criticism of Obama in recent weeks, and Obama has accused the voluble populist of "exporting terrorism" and being a "destructive force in the region."
In a column published in 28 Venezuelan newspapers Sunday, Chavez said he is willing to work toward improving relations with the U.S. but said Washington should "open its fists" first.
As for Osama bin Laden, he is giving Obama no breaks. In an audiotape that surfaced after Obama's election, bin Laden vowed to open "new fronts" against the United States, practically daring Obama to continue the wars in Iraq and Afghanistan.
"If he withdraws from the war, it is a military defeat. If he continues, he drowns in economic crisis," bin Laden said.
Meanwhile, he said, al-Qaida was prepared to fight "for seven more years, and seven more after that, and then seven more."
