The easing of lockdowns and a healthy run of economic data have provided a boost to world markets in recent weeks.
Analysts warned that while the employment data were good, jobless claims were still elevated.
Hong Kong led the gains on reopening after a one-day break, despite concerns about a new security law imposed on the city by China.
Markets have surged about a fifth from their March lows, thanks to the easing of lockdowns.
Worries about second-wave states appear now to be last week's news.
Several countries continued to loosen up, including in Britain where pubs, restaurants, hotels and cinemas were told they could open again from July 4.
So far this year, the auto market has shrank 41.5 percent due to the impact of coronavirus lockdowns.
In Europe, investors were underwhelmed by the Bank of England's latest monetary policy decisions, which undermined the pound and London stocks.
Eyes are on Beijing, which has imposed new lockdowns, reclosed schools and banned flights again.
Asian investors leary as signs of recovery offset by fears of second wave.
Beijing carried out mass testing after 36 of China's 57 new cases were linked to a wholesale food market.
The optimism on trading floors was shattered when Federal Reserve boss Jerome Powell signalled the world's top economy would take some time to bounce back.
While traders continue to buy into world markets on hopes the economy will bounce back, analysts urged some caution.
While tensions between China and the US continue to play in the background, the general mood at the start of June has been upbeat.
Governments in Europe and Asia have become confident enough to lift containment measures.
Global markets have been on a roll for weeks with countries slowly reopening after shutdowns and trillions of dollars in stimulus and central bank support,
Violent anti-racism protests across the US have fuelled worries of a pick-up in COVID-19 infections and more pain for the world's top economy.
Despite the threat of another trade war, investors are focusing on the easing of lockdowns around the world.
The US announced sanctions against a Chinese government institute and eight companies for human rights violations.
The World Bank warned the crisis could leave about 60 million in extreme poverty, adding that it saw the global economy contracting five percent this year.
US biotech firm Moderna reported "positive interim" results in early testing of a vaccine candidate.
The gains come despite a flurry of downbeat economic data, including Monday's news that Japan had fallen into its first recession since 2015.
The downbeat mood was compounded by another spike in US jobless claims and overshadowed news several countries were easing strict lockdown measures.
Federal Reserve officials also warned about the long-term financial impact of an extended shutdown.
Wuhan reported six new infections in two days and South Korea announced its biggest spike in new cases for more than a month.
There are concerns of a second wave hitting South Korea and China, which had been slowly reopening their economies.
Traders remain buoyed, as stimulus and central bank backstopping measures along with easing China US-tensions are providing much-needed reassurance.
With countries from Asia-Pacific to Europe and US states reopening their shattered economies, global equities have enjoyed a strong revival.
The easing of restrictions was providing a much-needed boost to oil markets.
Trump suggested he could lump new tariffs on China over its handling of the virus outbreak.