Asian equity indices mostly slid by the close, while in Europe the push lower was driven by Frankfurt with a fall of 2% around midday.
Paris shed nearly 2% as share prices of European automakers fell sharply.
Having launched tariffs on US neighbours and China at the weekend, Trump said he would impose levies on the EU.
He added that non-EU member Britain would likely avoid immediate tariffs, helping the London stock market to limit losses.
“Investors fear that this trade war will result in a significant deterioration in the global economy,” said John Plassard, investment specialist at Swiss asset manager Mirabaud.
There was a sharp selloff across the cryptocurrency sector, with bitcoin slumping around 7%.
Oil prices rallied, however, as Trump’s tariffs on Canada and Mexico include the commodity.
Haven investment gold slipped, having hit a fresh record above US$2,800 an ounce last week.
Trump’s action sees 25% levies on imports from Canada and Mexico and 10% duties on Chinese goods.
Analysts at Oxford Economics said the tariffs could see Mexican inflation surge to 6% annually, from 4.2% in December. The peso sank versus the dollar today.
Chief EY economist Gregory Daco said Canada’s economy could shrink 2.7% this year and 4.3% next year.
Canada said it would file a World Trade Organization claim against the US, while Mexican president Claudia Sheinbaum announced that retaliatory tariffs would be imposed on US products.
China’s trade ministry said Beijing would take “corresponding countermeasures”.
As for the EU, the bloc must show its muscle in the face of threatened tariffs, French president Emmanuel Macron warned.
“If we are attacked in terms of trade, Europe — as a true power — will have to stand up for itself and therefore react,” Macron said as he arrived for leaders’ talks in Brussels.
In Asia, the Year of the Snake started with a nasty bite for stock markets.
Tokyo, Seoul and Jakarta each shed more than 2% while Sydney, Bangkok and Wellington were each off more than 1%.
Singapore and India also fell, while Hong Kong gave up early deep losses to end only marginally down.
Shanghai remained closed for a holiday.
Taipei plunged more than 3%, with chip titan and market-heavyweight TSMC diving 5.7% on the first day of trade since China’s DeepSeek unveiled a cheaper artificial intelligence model rivalling those of US tech giants.