This is a testing time for America’s friends around the world. From Canberra to Copenhagen, governments are crossing their fingers that President-elect Donald Trump did not mean everything he said on the campaign trail.

If he did, their world is about to get a lot more dangerous.

Since the end of World War II, a network of US-focused military alliances has underpinned global peace and security. Mr. Trump has repeatedly questioned the value of those alliances, suggesting that America does not need them. Or at least that Washington should charge more money for them.

So, what’s at stake?

Q: What does Mr. Trump intend for NATO?

He has certainly spoken in surprisingly cavalier tones about the alliance, the bedrock of trans-Atlantic security. In April he complained that European allies are “ripping off the United States. Either they pay up … or they have to get out. And if it breaks up NATO, it breaks up NATO.”

Trump is right that Washington’s European allies are by no means paying their way on defense matters. US presidents up to and including Barack Obama have complained time and again of “free riders.” Only four of America’s 27 NATO allies have met the alliance target of devoting 2 percent of their GDP to defense, though European defense spending is now rising.

But “NATO is not a charity,” says Ruud van Dijk, a security analyst at the University of Amsterdam. It serves American purposes as much as European ones. “What does it mean for American interests, Trump’s top priority, if there is conflict in Europe?” he asks.

Trump has also repeatedly called NATO “obsolete.” Last April, he said the alliance should “upgrade” its “outdated mission … to confront our shared challenges including migration and Islamic terrorism.”

But NATO has participated in the 10-year war in Afghanistan – US allies provided one-third of the troops – in a major anti-terrorism effort. British, French, Dutch, and Danish planes are also currently flying combat missions over Syria and Iraq, targeting Islamic State fighters.

And NATO members have put together a naval force to support European Union frontier guards in their bid to stem illegal immigration in the Aegean.

Q. What about Russia?

Immigration and terrorism are not the only issues on NATO's watchlist – old-style threats persist, too. The three Baltic nations that were once part of the Soviet Union and are now NATO members – Lithuania, Estonia, and Latvia – are especially nervous about Moscow’s intentions, having watched Russia annex Crimea, a part of Ukraine.

Trump has indicated he might acquiesce in Crimea’s fate, and called for improved relations with Moscow.

The biggest threat to NATO, says Henrik Breitenbauch, head of the Center for Military Studies at Copenhagen University, is “Trump’s perception of what [Russian President Vladimir] Putin is up to.”

While most Western intelligence agencies warn that Mr. Putin's goal is to divide the West and undermine NATO’s collective defense structure, the incoming US president seems less suspicious, and less solicitous of that structure. He has suggested, for example, that he would examine the level of the Baltic states’ defense spending before deciding whether to defend them.

In fact, Estonia’s defense spending has already reached the required 2 percent of GDP, and its two neighbors are on track to hit that target in 2018.

But that is beside the point, says NATO Secretary-General Jens Stoltenberg. Under Article 5 of NATO’s 1949 founding treaty “all allies have made a solemn commitment to defend each other, and this is something that is absolute and unconditioned,” he said last week.

In fact, only a single NATO member has ever benefited from Article 5, treating an attack on one as an attack on all of them. That was the United States, after 9/11.

Q. Where does Trump stand regarding America’s allies in Asia?

Both North Korea and China are expected to test Trump’s strategic intentions in Asia with provocative moves early in his term. And Trump said a lot of things on the campaign trail that worried observers in the capitals of two key US allies, Japan and South Korea.

But none of them clarified a key question: Is he an isolationist, or is he a China hawk?

Some signs suggest that Trump is ready to abandon Asia to its fate. On the campaign trail, Trump threatened to pull American forces out of South Korea unless Seoul pays more for their upkeep. Under a deal reached two years ago, the South Korean government contributes about 40 percent of US military costs in the country.

But when Trump talked to South Korean President Park Gyeung-hye on the phone last Thursday, South Korean officials reported, he said he agreed “100 percent” with Seoul on the need to maintain the security alliance.

Japanese Premier Shinzo Abe, due to meet Trump in New York this week, will be hoping for similar reassurance. The US president-elect had said that unless Tokyo paid more for its defense, it should go it alone and perhaps build a nuclear weapon to protect itself.

If Japan, the only country to suffer a nuclear attack, tried to build such a device of its own, it would almost certainly provoke insurmountable opposition among the Japanese people – and a destabilizing arms race in Asia.

Trump has insisted that he will not allow North Korea to develop a usable nuclear weapon, but has not yet said how he plans to achieve that goal, other than proposing to meet North Korean leader Kim Jong-un for talks.

Q: Is anyone else in that part of the world worried?

In Australia, one of America’s most distant but most loyal allies, Trump’s ascension has sparked unprecedented doubts about the future of a defense partnership with the United States that has undergirded the country’s peace and prosperity since the end of World War II.

With the signing of the ANZUS Treaty in 1951, Washington guaranteed Australia’s security as it developed into one of the richest societies on earth. In return, Australian troops have fought alongside American forces in every war that Washington has fought.

After an isolationist campaign, Mr. Trump’s election has thrown long-held assumptions about Australia’s security into question just as China flexes its military muscle more assertively.

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“Australia’s approach to the rise of China has been to build a deeper and broader alliance with the United States,” says Dr. Peter Dean, a defense analyst at Australian National University. “If we become an independent state, that is going to cost us potentially double our defense spending to make ourselves somewhat secure.”

• John Power in Melbourne contributed to this article.