In his inaugural address, President Barack Obama signaled conciliation to America's foes by using the metaphor of an outstretched hand to an unclenched fist.
Already, there are signs that some of those foes 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 demand in return from the untested American statesman, 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 really 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, the 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 nations are deep _ U.S. suspicions about Iran's nuclear program, President Mahmoud Ahmadinejad's threats to annihilate Israel, and Tehran's support for militants in Iraq _ and 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 experts feel the Islamic nation is committed to developing nuclear weapons, whatever the cost.
Obama may think "he can convince Iran to give up its nuclear program, but this is a red line for Iran," said Saudi political analyst Khaled al-Dakhil.
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 maneuver.
Syria, which has teetered between pariah and potential peace partner, has indicated that it seeks no further quarrel with Washington, even saying it would like the new administration to mediate stalled Syrian-Israeli peace talks.
If Obama helps Syria in from the cold, analysts say it could have far-reaching benefits. Syria has great influence over two of Israel's main enemies _ Hezbollah in neighboring Lebanon, and Hamas, whose leaders live in Damascus. Assad could also help pressure Syrian ally Iran to take a more measured 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 nonstarter to Israelis, but still, his conciliatory language marked a striking turnaround from the terrorist-sponsoring Gadhafi 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 isolated, nuclear-armed regime said the country 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 missile defense plan and support for Ukraine and Georgia joining NATO.
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, the chairman of the presidium of Russia's Council on Foreign and Defense 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 hard-line tactics in shutting off natural gas to Europe during a dispute with Ukraine.
Yevgeny Volk, coordinator of the Heritage Foundation's Moscow office, warned Obama to be ready for a Russian test, recalling that in 1961, "Russia tried to test in the same manner John Fitzgerald Kennedy, who the Russian leadership also believed to be a relatively young and soft leader."
In America's own hemisphere, the greatest hope for diplomatic progress lies with Cuba, where Fidel Castro and his brother-successor, Raul Castro, 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.
"It would be very difficult for Chavez to go back on years of portraying the U.S. as the root of all evil in Latin America and as an impediment to his own vision for the region," said Peter DeShazo, director of the Americas program at the Center for Strategic and International Studies in Washington.
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."
Editor's Note: Paul Haven is the Associated Press bureau chief in Madrid. AP writers Rachel Jones in Caracas, Niko Price in Havana and Douglas Birch in Moscow contributed to this report.