Global stock markets have had a bumpy ride in recent weeks
US shares clawed back some of Monday's heavy losses, after President George W Bush renewed calls for Congress to back a $700bn (£380bn) banking rescue plan.
With Wall Street having seen record falls after politicians blocked the deal, the Dow Jones index rose 485 points or 4.7% by the close of trading.
Analysts said investors were hopeful a new deal could be agreed this week.
It has been announced that the US Senate will vote on a revised version of the plan on Wednesday evening.
The gains on Wall Street encouraged strong performances in Europe, with the FTSE 100 index closing 1.7% ahead.
Mr Bush warned that if an agreement was not reached, the US economy faces "painful and lasting damage".
"We are in an urgent situation and the consequences will grow worse each day if we do not act," Mr Bush said at the White House.
"It matters little what path a bill takes to become law. What matters is that we get a law. We're at a critical moment in our economy."
While the Dow Jones reached 10,850.7 points, the other main Wall Street index - the Nasdaq - advanced 5% to 2,082.3 points, having shed 9.1% in the previous session.
On the back of the strong gains, European shares closed ahead.
The UK's FTSE 100 added 83.4 points to 4,902.5, France's key index added 2% to 4,032.1 points while Germany's Dax ended 0.4% higher at 5,831 points.
Earlier, Japan's Nikkei index ended Tuesday down 4.1%, while Hong Kong's Hang Seng rose 0.8%.
Oil, which had fallen by about $10 a barrel on Monday, also rallied.
Expectations that politicians would pass some form of financial stability plan gave investors hope that demand for energy would increase, observers said.
US light, sweet crude rose by $4.27 to settle at $100.64 a barrel while London Brent crude added $4.19 to $98.17 a barrel.
Analyst Richard Hunter, head of equities at Hargreaves Lansdown stockbrokers, said European investors were hopeful the US would eventually pass the bail-out plan.
"This deal is not dead in the water and there are hopes that when Congress reconvenes it could still go through," he said.
There were a number of other key financial events on Tuesday:
- In Russia, trading was temporarily suspended on the country's two main stock markets
- In the Republic of Ireland, the government announced that all bank deposits would be guaranteed for the next two years
- European bank Dexia has received a state bail-out, costing the Belgian, French and Luxembourg governments a combined 6.4bn euros ($9.2bn; £5bn)
- Several banking stocks fell on the FTSE 100 index, with HBOS down 13.8%, Barclays losing 2% and Royal Bank of Scotland slipping 1.1%
Day of turmoil
The US rescue plan, a result of tense talks over several days between the government and lawmakers, was rejected by 228 to 205 votes in the House of Representatives.
HAVE YOUR SAY
I know we need a strong financial sector, but where is the talk of structural change that's going to prevent recurrence?
Neil, California, US
About two-thirds of Republican lawmakers refused to back the rescue package, as well as 95 Democrats.
Congress will not meet again until Thursday - after a break for the Jewish New Year - with another vote unlikely before the weekend, the BBC's Jonathan Beale in Washington says.
The House's rejection of the bail-out plan came after a day of turmoil in the US and Europe, with Wachovia, the fourth-largest US bank, being bought by larger rival Citigroup.
Monday also saw the partial nationalisation of Benelux banking giant Fortis by three governments, and UK lender Bradford & Bingley was taken into state ownership.