In London, Paris and Frankfurt stocks traded about 3% up, following a 7.7% rise for New York’s Dow Jones index.
“Equities are still racing higher, as the news from key countries like Spain and Italy remains positive,” said IG analyst Chris Beauchamp.
The pound and oil prices also edged higher.
In the UK, travel, tourism and entertainment companies were among the biggest gainers. Carnival jumped 23%, while Easyjet rose 20%. British Airways’ owner IAG rose 12%.
In early afternoon trading, 17 companies in the FTSE 100 were trading more than 10% up.
The FTSE 250, considered to be more representative of the UK economy, was trading more than 5% higher.
Meanwhile, oil stocks also climbed slightly as traders expected an end to a price war between Russia and Saudi Arabia.
Investor sentiment was buoyed by news that the death toll in Spain, which has been badly affected by the coronavirus pandemic, had fallen for a fourth consecutive day, in a sign that the country may have passed its peak.
There was also a slowdown of new infections in Italy, which has also been ravaged by the virus. News that some countries, including Austria and Denmark, have made small steps to relax their lockdowns also helped sentiment.
“Investors are reacting to indications that lockdown measures in the UK, US and Europe are beginning to ‘flatten the curve’ of coronavirus infections and fatalities,” said Russ Mould, investment director at AJ Bell.
However, he added: “The market’s relief is only likely to last so long, and attention will soon turn to how countries intend to exit the current containment measures which have in effect hit the pause button on the global economy.”