‘America is back.’ What that means to the world.

“Welcome back America!”

The celebratory tweet from Paris Mayor Anne Hidalgo came just minutes after the U.S. television networks had called the presidential election in favor of Joe Biden last weekend, and her response has been echoed in the days since by many U.S. allies around the globe.

The underlying message: In an increasingly fractious world there is a strong desire to feel America’s presence. Especially in the face of borderless challenges like the COVID-19 pandemic and climate change, hopes are high that the most powerful democracy on earth will reengage internationally, move away from President Donald Trump’s highly personalized “America First” approach, and resume something more like its traditional leadership role on the world stage.

That view is particularly prevalent among America’s traditional European allies, who were deeply concerned by the prospect of an additional four years of Mr. Trump’s policies. They voiced alarm at Mr. Trump’s preemptive claim of victory as the polls closed, and this week, the leaders of France, Germany, and Britain shrugged off the president’s continuing refusal to accept the election result to engage personally with President-elect Biden.

They, along with America’s main partners in Asia, are not taking this path primarily because they expect any particular policy changes under a Biden administration.

Rather, they share a broader hope, less about American policy than American presence – both in multilateral institutions, and as a source of democratic ballast in an unsteady world. The absence of this kind of engagement, in their view, has emboldened autocratic regimes, hamstrung advocates of human rights, and left the field open for major powers like China and Russia to become more assertive.

Few expect as dramatic a change as was suggested by former Polish Prime Minister Donald Tusk just after Mr. Biden’s victory. “Thank you, Joe,” he tweeted, adding that “Trump’s defeat can be the beginning of the end of the triumph of far-right populism.” Still, many observers share the view that autocratic or authoritarian governments will feel more constrained, and such regimes, themselves, do appear to have sensed a change in the wind.

As is often the case in international affairs, it’s been important since the election to watch for the old Sherlock Holmes clue: the dog that doesn’t bark. While leaders of the world’s democracies were sending messages of congratulations to the president-elect, a number of other countries initially held back.

Among them were the self-proclaimed “illiberal democracies” of Hungary and Poland. Turkey and Saudi Arabia as well. And the few leaders delaying even further included China’s Xi Jinping, Russia’s Vladimir Putin, and Jair Bolsonaro, the far-right populist president of Brazil.

The full policy impact of a Biden administration will become clear only further down the road, and even those allies quickest to welcome the election result are assuming that the new president will focus first on domestic challenges: the headlong spread of COVID-19 and its hugely damaging economic effects, and healing a deeply divided America.

But they do view one global campaign commitment as enormously important: his pledge to reverse Mr. Trump’s withdrawal from the 2015 Paris Agreement on the day he takes office. That’s not just because of its symbolic significance, as a high-profile example of America’s reentry into multilateral diplomacy. It’s also a practical matter. It will not be possible to speed up international efforts to contain global warming unless the U.S. plays a leading role.

Among those who this week strongly welcomed that prospect, in fact, was the European leader personally closest to Mr. Trump, British Prime Minister Boris Johnson. He is due to host the follow-up conference to the Paris accord in Glasgow next year.

America’s partners are also anticipating U.S. efforts to repair two transatlantic relationships that have frayed badly during Mr. Trump’s presidency – ties with NATO and with the European Union.

They also expect Mr. Biden to reverse the Trump administration’s withdrawal from the World Health Organization, as well as to seek a shared approach with allies in response to Chinese leader Xi Jinping’s tightening political control at home and moves to extend China’s economic, political, and security influence abroad.

But even as U.S. allies nurture these hopes, one key question remains: How comprehensively would, or could, a Biden administration reverse the full diplomatic impact of the past four years? They are keenly aware that nearly half of Americans voted to reelect Mr. Trump. It remains to be seen whether the new president will command a majority in the Senate. And his own Democratic Party harbors opponents of free trade and overseas security commitments.

Beyond this, Mr. Trump’s public criticism of long-standing allies, his denigration of both NATO and the European Union, and his sheer unpredictability have seriously undermined transatlantic trust.

French President Emmanuel Macron, in particular, has been arguing that the EU needs to strengthen its own voice and influence independent of its alliance with Washington. German Chancellor Angela Merkel, long reluctant to embrace this kind of vision, has become more open in recent months to the need for closer economic, financial, and political cohesion inside the union.

In remarks this week making it clear she was looking forward to working with Mr. Biden, Ms. Merkel added a note of caution. “We Germans, and Europeans, understand that we must take on more responsibilities in this partnership in the 21^st century. America remains our most important ally, but it rightfully expects more of us to guarantee our own security and defend our values around the world.”

And in the immediate aftermath of the U.S. election, European Commission President Ursula von der Leyen predicted that even under a Biden presidency, “we won’t pick up things where we left off in 2016. The world has changed.”