Hopes that Mr Trump will introduce a pro-business agenda to stimulate US economic growth helped to blunt concerns about his surprise win.
The main stock markets in London, Frankfurt and Paris hovered about 1% higher for most of morning trading.
Earlier, there were big jumps in Asia, where Japan's Nikkei 225 soared 6.7%.
In the US, the futures index suggested that Wall Street would open later almost 1% up.
