• Adding to the broadly positive outlook is optimism about progress on a vaccine, which would allow the shattered global economy to start bouncing back.
• London's rally came despite a government crisis in Britain where the number of deaths "involving" the coronavirus has meanwhile risen to 46,000.
• Tokyo - Nikkei 225: UP 2.6 percent at 21,271.17 (close).

While countries including Brazil, Chile and Russia are enduring rising death and infection tolls from COVID-19, an increasing number of governments are seeing their figures tail off enough to attempt to get society back to some form of normal.

"Once again the markets embraced an optimistic outlook... setting aside fears over the long-term economic impact of the pandemic and the ever-growing tensions between the US and China to focus on another round of global easing measures," said Connor Campbell, analyst at trading group Spreadex.

Adding to the broadly positive outlook is optimism about progress on a vaccine, which would allow the shattered global economy to start bouncing back.

But Chris Iggo, at AXA Investment Managers, warned: "That does not mean we should ignore the risk of second waves, prolonged weak growth and geopolitical issues."

Tokyo rose more than two percent Tuesday and Sydney jumped nearly three percent, while Shanghai, Taipei, Seoul, Jakarta, Bangkok and Wellington closed up more than one percent.

Hong Kong rose 1.9 percent, as city leader Carrie Lam sought to reassure investors, saying fears its business-friendly freedoms were at risk from a planned Chinese national security law were "totally groundless."

Lam also reasserted a pledge to "the independence of the judiciary, the various rights and freedoms enjoyed by people".

London, playing catch-up with strong eurozone gains Monday, jumped more than two percent in early trade but slipped back, while Paris and Frankfurt were up more than one percent in midday deals.

London's rally came despite a government crisis in Britain where the number of deaths "involving" the coronavirus has meanwhile risen to 46,000.

In Asia, Beijing's decision to push its national security law on Friday sent the Hang Seng tumbling more than five percent and ramped up already high tensions with Donald Trump, who has continually hit out at China for its alleged role in the spread of the coronavirus.

But National Australia Bank's Rodrigo Catril said that while their standoff is simmering, "equity investors appear more interested in the prospect of economies reopening around the globe".

Elsewhere Tuesday, oil markets pushed on with their recovery, having suffered a spectacularly bad April when WTI crashed below zero.

The reopening of economies and a massive cut in output by some of the world's top producers has helped the US benchmark virtually double in value this month.

Key figures at around 1045 GMT

London - FTSE 100: UP 1.3 percent at 6,068.84 points

Frankfurt - DAX 30: UP 0.9 percent at 11,487.82

Paris - CAC 40: UP 1.3 percent at 4,597.52

EURO STOXX 50: UP 0.7 percent at 2,993.23

Tokyo - Nikkei 225: UP 2.6 percent at 21,271.17 (close)

Hong Kong - Hang Seng: UP 1.9 percent at 23,384.66 (close)

Shanghai - Composite: UP 1.0 percent at 2,846.55 (close)

New York - Dow: Closed for public holiday

West Texas Intermediate: UP 2.4 percent at $34.06 per barrel

Brent North Sea crude: UP 1.4 percent at $36.01 per barrel

Euro/dollar: UP at $1.0957 from $1.0896 at 2100 GMT Friday

Dollar/yen: DOWN at 107.67 yen from 107.71 yen

Pound/dollar: UP at $1.2309 from $1.2195

Euro/pound: DOWN at 88.99 pence from 89.35 pence