By Agence France-Presse
US stocks showed little life early in the day but rose after Clinton publicly conceded her loss and called for an orderly transfer of power. The Dow Jones Industrial Average rose 1.4 percent, while the S&P 500 and Nasdaq each finished 1.1 percent higher.
Stock futures and the dollar fell sharply overnight as word of Trump's victory spread. Asian markets dove at the start of the day, with Tokyo sinking on fears of the New York billionaire's untested policies. European markets also fell about 2 percent at the open but later rallied.
At the end of the European day, London stood 1.0 percent higher, Frankfurt added 1.6 percent and Paris gained 1.5 percent.
"Certainly a much unexpected move for the market compared to the rollercoaster we've seen in the last 12 or 18 hours," said David Levy of Republic Wealth Advisors.
"The market is digesting who the winners and losers will be and we'll continue to do so in the coming days but for now the initial reaction is positive."
'Leap into the unknown'
The US election outcome has drawn direct comparisons with Britain's shock Brexit vote in June to leave the European Union.
"Of course, just as Britain has not yet Brexited, America has not officially entered the era of Trump," Spreadex analyst Connor Campbell said.
Rebecca O'Keeffe, head of investment at stockbroker Interactive Investor, said there are "significant" worries over Trump's policies.
"Trump is a huge leap into the unknown," she told AFP.
But CMC Markets analyst Jasper Lawler noted, "things are never black and white and Donald Trump can be a positive force, or at least a less negative force than Hillary Clinton, for some sectors of the stock market."
Pharmaceutical stocks surged as Clinton's campaign for cheaper drugs was off the table after her defeat, with Pfizer jumping more than 7 percent, and Eli Lilly, Sanofi, Bayer and others all sharply higher.
Defense stocks also rose, on expectations of higher US military spending, with Lockheed Martin gaining nearly 6 percent and Northrup Grummon up 5.4 percent.
Banks also saw their shares gain on expectations of reduced regulation, with JPMorgan Chase rising 4.6 percent, Goldman Sachs gaining 5.9 percent and Wells Fargo up 5.4 percent.
But Spain's second-biggest bank, BBVA, fell nearly 6 percent, punished by investors for its high exposure to the Mexican market.
The Mexican peso -- which has been battered by Trump's anti-Mexican rhetoric including promising construction of a border wall -- continued its declines. The dollar recovered from early losses and strengthened slightly against the euro.
In Asia, Tokyo stocks collapsed 5.4 percent and Hong fell 2.2 percent.
Moscow however rose almost 2 percent on hopes of improving US relations as President Vladimir Putin congratulated Trump.
Key figures around 2100 GMT
New York - Dow: UP 1.4 percent at 18,589.84 (close)
New York - S&P 500: UP 1.1 percent at 2,163.26 (close)
New York - Nasdaq: UP 1.1 percent at 5,251.07 (close)
London - FTSE 100: UP 1.0 percent at 6,911.84 (close)
Frankfurt - DAX 30: UP 1.6 percent at 10,646.01 (close)
Paris - CAC 40: UP 1.5 percent at 4,543.48 (close)
EURO STOXX 50: UP 1.1 percent at 3,056.29 (close)
Tokyo - Nikkei 225: DOWN 5.4 percent at 16,251.54 (close)
Hong Kong - Hang Seng: DOWN 2.2 percent at 22,415.19 (close)
Shanghai - Composite: DOWN 0.6 percent at 3,128.37 (close)
Dollar/Mexican peso: UP at 19.82 pesos from 18.37 pesos
Pound/dollar: UP at $1.2424 from $1.2378 Tuesday
Euro/dollar: DOWN at $1.0914 from $1.1020
Dollar/yen: DOWN at 105.72 yen from 105.14 yen
Oil - West Texas Intermediate: UP 29 cents at $45.27 per barrel
Oil - Brent North Sea: UP 32 cents at $46.36