London's FTSE 100 index rose 1.4%, Paris's Cac gained 1.1%, and Frankfurt's Dax was up 1.2%.
The Federal Reserve said it would not scale back its bond-buying programme until the US economy had improved further.
The bank also cut its growth forecast.
The Fed now estimates that the US economy will grow by between 2.0% and 2.3% this year. That compares to a previous estimate made in June of between 2.3% and 2.6%.
Relief
Markets rose on the news because it means the cheap money resulting from the US stimulus plan will continue.
Gold rose 4% - its biggest one-day jump in four years - following the Fed's decision, and was trading at $1,366 per ounce on Thursday morning.
Gold mining stocks rose sharply, with Randgold Resources up 9.4% and Fresnillo up 6.4%.
In Asia, Japan's Nikkei 225 closed up 1.5% and Hong Kong's Hang Seng ended 1.7% higher.
Australia's ASX 200 rose 1.2% to a five-year high after the announcement.
In the US, the three main stock indexes closed at record highs on Wednesday.
'Elevated' unemployment
Many investors had speculated that the Federal Reserve would begin reducing its monthly bond-buying plan this month.
But in a statement released after its two-day policy meeting, the Fed said there was no fixed timetable for it to begin scaling back - or "tapering" - its stimulus.
The central bank said it was taking a more cautious stance because of an "elevated" unemployment rate and concerns about the US economic recovery.
Its stimulus programme was designed to keep interest rates low, and thereby money cheap in order to encourage borrowing and investment.
Emerging markets were galvanised by the Fed's decision not to start scaling back its stimulus programme.
Investors had sought out higher returns in shares and other assets in faster-growing economies such as Brazil, India and Indonesia.
These had seen strong growth as a result of the Fed's attempts to boost the US economy, but conversely had fallen back sharply at the prospect of a return to higher borrowing costs in the US.
'Radical'
However, following the Fed's latest announcement, India's main share index rose 3%, Indonesia's climbed 4.7% and Turkey's stock market was up by 6%.
Investment bank Societe Generale said in a research note to clients that the outlook for emerging markets had greatly improved.
"The Fed signalling has now changed our perspective on the market in a radical way," the bank said.
"Against this backdrop - a true game-changer - it is no longer appropriate to hold a strategically bearish view on global emerging markets."
Reports that Janet Yellen was the front runner to take over from Ben Bernanke as head of the US Fed added to the view that a tightening in US monetary policy would not happen soon, as she is considered to see the risks of poor economic growth as bigger than that of rising inflation.
Murat Toprak, emerging FX strategist at HSBC said: "We can assume this rally is going to last for some time, probably at least a few weeks.
"There is a reassessment by the market of monetary policy changes going forward."
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