President-elect Joe Biden’s vow to restore America’s leadership in the world will be swiftly tested by resurgent adversaries, rudderless institutions and the gravest global health crisis in decades.The big picture: Biden will face a familiar antagonist in Moscow, a stronger and more assertive China, a nuclear-armed North Korea, and an ongoing war in Afghanistan. That's not to mention a pandemic that’s ravaged the world and darkened the global economic outlook.Biden’s approach to all of those challenges starts with revitalizing America’s alliances and restoring its credibility. His message is that “America is back” and “America first” is over. Biden plans to quickly re-enter the Paris climate accord, recommit to NATO and re-engage with the World Health Organization.He plans to put human rights back at the heart of U.S. foreign policy, and he has proposed a summit of the world’s democracies.But Biden will have to prioritize the crises at home even as he aims to restore America's standing abroad.Biden’s top domestic focus — containing the virus and its economic fallout — may also be his greatest foreign policy challenge.Distributing vaccines globally will be particularly important. Competition over doses will drive resentment and prolong the pandemic, even if Americans get access relatively soon.Multiple members of Biden's coronavirus task force have been sounding that alarm, portending a different approach from Biden after Trump shunned the WHO’s vaccine initiative.Salvaging the Iran nuclear deal is near the top of Biden’s priority list. That would require lifting sanctions if Iran — which now has 12 times the amount of low-enriched uranium permitted under the deal — returns to compliance.Iranian leaders have suggested they’d be willing to return to the old deal, but they dismiss talk of the broader follow-on agreement Biden envisions.The Trump administration, meanwhile, is attempting to block the path back to the deal by piling on new non-nuclear sanctions that Biden might find politically tricky to lift.Biden will have to move quickly to extend the New START nuclear treaty with Russia, which is set to expire 15 days after he takes office. But he’s also vowed to confront Vladimir Putin over election interference and “impose real costs on Russia for its violations of international norms.”Bill Burns, president of the Carnegie Endowment and a former ambassador to Moscow, told Axios it will be important for Biden to set the terms of the relationship early: “We’re going to be operating within a pretty narrow band of possibilities in dealing with Vladimir Putin’s Russia — from the sharply competitive to the pretty nastily adversarial."North Korea may force itself onto Biden's agenda soon after he enters the White House. The regime has a history of testing incoming U.S. administrations.Kim Jong-un's nuclear program has advanced significantly in the last four years, even as he has temporarily suspended testing.The leaders aren’t starting on great terms. Biden called Kim a “thug” during the campaign, while North Korea labeled Biden a “rabid dog.”Biden will also inherit a war in Afghanistan that he and Barack Obama vowed to end more than a decade ago.U.S. troops are withdrawing from the country even as peace talks between the Taliban and Afghan government stall. Biden plans to continue to pull troops out, but he may leave a counterterrorism force behind.