China and the U.S. are working hard to end their trade dispute:
That's one factor driving the gains.
But for Asia, e-commerce giant Alibaba boosted activity ...
As its shares started trading in Hong Kong, after the world's largest share sale of the year.
That helped markets across the region - and globally the MSCI 49-country main world share index is now less than 1% from an all-time high.
Monday's U.S. shares were already there:
All three major indices hit new records on trade hopes.
And after news of Charles Schwab's 26 billion dollar deal to buy broker TD Ameritrade hit the wires, Europe's big deal is LVMH buying jeweller Tiffany for 16 billion.
Fiat Chrysler and Peugeot-owner PSA, meanwhile, have told their employees they'll sign a binding merger agreement within weeks.
But bourses slipped into the red on profit-taking early on Tuesday.
And the FTSE was weighed down by catering firm Compass Group.
It sees, it said, deteriorating confidence in Europe hurting volumes and margins.
Though on Tuesday - with a 7% loss - its shares were taking the pain.