The Dow Jones industrial average rose 135 points to 13,572 in the first hour of trading.
The market was already higher in the opening minutes of trading, then jumped at 10 a.m. after the Institute for Supply Management reported that U.S. manufacturing grew in September for the first time in four months.
The Standard & Poor's 500 index was up 15 points to 1,456. The Nasdaq composite rose 29 points to 3,145.
Monday is the first day of trading of the fourth quarter. U.S. indices fell on Friday, for the fifth day out of six.
Quincy Krosby, market strategist at Prudential Financial, said investors believe that the news about the economy has stopped getting worse. Besides the U.S. manufacturing news on Monday, she noted that recent data from China suggests that manufacturing has improved there as well.
"The numbers were still weak, but they were not as bad as before," Krosby said. "So that was a positive backdrop for the market."
Markets around Europe rose. An audit of 14 Spanish banks showed the lenders need an extra $77.6 billion in capital. That's roughly what was expected, and well within the amount Madrid can get from fellow European countries.
A slight improvement in a survey of the euro zone's manufacturing sector also helped.
However, credit rating agency Moody's might downgrade Spain's debt to junk status this week. That's likely to limit enthusiasm in Europe until the Moody's decision is known.
Germany's DAX stock index rose 1.2 percent, France's CAC-40 was up 1.7 percent, and Britain's FTSE 100 rose 1.3 percent. Spain's Ibex was up 0.8 percent.
Markets in China, Hong Kong and South Korea were closed for holidays. Japan's benchmark Nikkei 225 index fell after a closely watched survey showed confidence in the economy weakening.
The euro rose to $1.291.
Yields on U.S. 10-year Treasury notes fell slightly to 1.635 percent.