France's CAC 40 added 1.5% in early trading to 4,443.47. Germany's DAX gained 2.5% to 10,063.41, while Britain's FTSE 100 jumped 2.2% to 5,685.74.

U.S. shares were set to drift higher, with Dow futures edging up 0.6% to 22,460.20. S&P 500 futures also rose, by 0.7% to 2,645.18.

Still, it was a welcome sign of resilience.

Japan's benchmark Nikkei 225 rose in morning trading but reversed course to dip nearly 0.9%, finishing at 18,917.01. Australia's S&P/ASX 200 also fell back, losing 2.0% to 5,076.80, while South Korea's Kospi picked up 2.2% to 1,754.64. Hong Kong's Hang Seng stood at 23,603.48, up 1.9% and the Shanghai Composite inched up 0.1% to 2,750.30.

India's Sensex jumped 3.6% to 29,467.39. Shares rose in Thailand, Indonesia and Singapore.

An official survey showed China's manufacturing rebounded in March as authorities relaxed anti-disease controls and allowed factories to reopen. But an industry group warned Tuesday that the economy has yet to fully recover.

The purchasing managers' index issued by the Chinese statistics bureau and an official industry group rose to 52 from February's record low of 35.7 on a 100-point scale on which numbers above 50 show activity increasing.

The overnight rally on Wall Street tacked more gains onto a recent upswing for the market, which is coming off the best week for the S&P 500 in 11 years, albeit after falling into bear market territory. Optimism is budding that the worst of the selling may be approaching, but markets around the world are still wary as leaders work to nurse their economies through the pandemic. The S&P 500 remains 22.4% below its record set last month, and oil tumbled to an 18-year low.

"Despite some respite for markets overnight, uncertainty remains as the spread of the COVID-19 virus continues," said Zhu Huani at Mizuho Bank, warning against too much optimism.

"Central banks and authorities continue to step up measures to support the economy."

Sentiment was brightened by news of potential developments in the fight against the coronavirus outbreak.

Johnson & Johnson leaped 8% after saying it expects to begin human clinical studies on a vaccine candidate for COVID-19 by September. Abbott Laboratories jumped 6.4% after saying it has a test that can detect the new coronavirus in as little as five minutes.

Stocks jumped last week after the Federal Reserve promised to buy as many Treasurys as it takes to get lending markets running smoothly and Capitol Hill reached a deal on a $2.2 trillion rescue package for the economy.

"The market wants to see everything line up, and last week everything lined up," said Nela Richardson, investment strategist at Edward Jones, referring to the unprecedented aid from the Fed and Congress.

Now, she said, President Donald Trump also appears to be in sync with health experts about the need to restrict the economy to slow the spread of the virus. Trump on Sunday extended social-distancing guidelines, which recommend against group gatherings larger than 10, through the end of April. Earlier, he had said he wanted the economy open by Easter.

"Now that message is in line," said Richardson. "All these things line up coming into this week, and that's why you saw strong performance last week continuing today."

Economists expect a number of weak reports on the economy to come in through the week. The lowlight will likely be Friday's jobs report, where economists expect to see the steepest drop in the nation's payrolls since the Great Recession.

The number of known infections around the world has topped 780,000, according to Johns Hopkins University. The United States has the highest number in the world, more than 160,000.

Most people who contract COVID-19 have mild or moderate symptoms, which can include fever and cough. But for others, especially older adults and people with existing health problems, the virus can cause pneumonia and require hospitalization.

More than 37,000 have died worldwide due to COVID-19, while more than 160,000 have recovered.

ENERGY: Benchmark U.S. crude added $1.10 to $21.19 a barrel. It fell 6.6% to $20.09 a barrel on Monday, after touching its lowest price since 2002. Oil started the year above $60 and has plunged on expectations that a weakened economy will burn less fuel. The world is awash in oil, meanwhile, as producers continue to pull more of it out of the ground.

Brent crude, the international standard, picked up 72 cents to $27.14 per barrel.

CURRENCIES: The dollar was trading at $108.34, up from $107.76 on Monday. The euro was little changed, at $1.0996 from $1.0995.