In London, the FTSE 100 index was up by 2.88% at close and in France and Germany the major indexes had risen by 4.09% and 3.39% respectively.
At close on Wall Street, the Dow Jones ended the day 2.94% ahead.
However, analysts said gains might not be sustainable, with many market participants still on holiday and low trading volumes.
“It is customary to greet the New Year with a surge of optimism,” said Stephen Lewis, an analyst at Monument Securities.
“Past cares are buried as eyes are raised to more distant horizons.”
The FTSE 100 had suffered its worst year on record in 2008 – a fall of 31.3%.
In Asia, South Korea’s main stock index closed up 2.9% at 1,157.40 points. Hong Kong’s Hang Seng index rose 4.6% and Indian shares climbed 0.6%.
Markets in Japan and China were closed for a public holiday.
At close in the US, the Nasdaq was also ahead, by 3.50%, and the S&P 500 was up 3.16%.
Global markets saw record falls in 2008 as the financial turmoil and economic slowdown ended the stock market boom.
Shanghai was one of the worst-hit major markets, ending the year 65% lower, which was also a record loss.
In New York, the Dow Jones lost almost 34% of its value in 2008, its worst year since 1931.
The year saw the credit crisis push several major economies into recession, with banks particularly badly hit – many requiring government bail-outs.
Whether the stock markets fall further in 2009 is a matter of debate.
Many investment strategists have written off any chance of a major rebound in at least the first six months of the new year, when company earnings could prove especially bleak.