New York extended its positive trajectory from the previous week, which had also pulled Asia higher.
In Europe, London and Frankfurt rose, but Paris sank.
"As we start a new week, the focus is once again on tariffs and earnings reports," said Kathleen Brooks, research director at trading group XTB.
Investors in US equities have been encouraged by forecast-beating results from major corporations, against only a modest uptick in inflation that suggested Trump's tariffs impact was not yet a worry.
But analysts warned the picture could change if Trump made good on his threat to slap higher tariffs on major US trading partners the European Union, Canada and Mexico.
Brooks and others stressed "the clock is ticking" towards August 1, when a bruising US-EU trade war could be unleashed.
Brussels has readied reprisals against a range of US imports -- including on Boeing planes and bourbon -- should no breakthrough come in its negotiations with Washington.
Trump has threatened 30-percent tariffs on EU goods, which would rise further if Brussels retaliated.
"The upcoming US tariff deadline, which is due to kick in a week this Friday, continues to cast a long shadow, particularly across the EU," said David Morrison, senior market analyst at Trade Nation.
US Commerce Secretary Howard Lutnick told CBS News over the weekend he was "confident" a trade deal would be reached with the EU.
But Jochen Stanzl, chief market analyst at CMC Markets, said that any agreement would likely be "only a framework deal... requiring further negotiations on the details".
"Realistically, there is a high probability that uncertainty will persist beyond August 1," he said.
That uncertainty will be part of the the European Central Bank's calculus as it meets this week. Expectations are for it to hold eurozone interest rates steady, pausing a long cycle of easing.
Asia's equities advance was led by Hong Kong and came after strong earnings from Taiwanese chip giant TSMC and news that US titan Nvidia will be allowed to export key semiconductors to China.
The yen strengthened against the dollar after Japanese Prime Minister Shigeru Ishiba vowed to stay on even after his ruling coalition lost its majority in the upper house in elections on Sunday.
Ishiba, too, is struggling to reach a trade deal with Trump, who has threatened tariffs of 25 percent on goods from Japan.
In company news, Jeep maker Stellantis said it suffered a massive, 2.3-billion-euro ($2.7-billion) net loss in the first half of this year, on the back of slumping North America sales and partly from "the early effects of US tariffs".
It shares, which have lost more than a third of their value since the start of the year, dipped early on Monday before reversing course and ending up.
Oil prices receded on worries of declining global trade.
But US Treasury Secretary Scott Bessent suggested the next round of US-China talks could include Chinese purchases of Russian and Iranian oil, which might pressure supply.
Trump has already warned he will impose tariffs on countries buying Russian oil if Moscow did not end its war on Ukraine, and Bessent said: "I would urge our European allies, who have talked a big game, to follow us if we implement these secondary tariffs."
- Key figures at around 1545 GMT -
New York - Dow: UP 0.5 percent at 44,572.12 points
New York - S&P 500: UP 0.6 percent at 6,334.34
New York - Nasdaq Composite: UP 0.7 percent at 20,048.39
London - FTSE 100: UP 0.2 percent at 9,012.99 (close)
Paris - CAC 40: DOWN 0.3 percent at 7,798.22 (close)
Frankfurt - DAX: UP 0.1 percent at 24,307.80 (close)
Hong Kong - Hang Seng Index: UP 0.7 percent at 24,994.14 (close)
Shanghai - Composite: UP 0.7 percent at 3,559.79 (close)
Tokyo - Nikkei 225: Closed for a holiday
Euro/dollar: UP at $1.1709 from $1.1627
Pound/dollar: UP at $1.3503 from $1.3414
Dollar/yen: DOWN at 147.28 yen from 148.73 yen on Friday
Euro/pound: UP at 86.73 pence from 86.67 pence
Brent North Sea Crude: DOWN 0.5 percent at $68.97 per barrel
West Texas Intermediate: DOWN 0.4 percent at $65.77 per barrel
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