@kees
Quote: Why should links always be underlined?

Like Clair and Miroslav pointed out already, it's for accessibility. Ask yourself if someone colorblind, or otherwise visually impaired, is able to find the links on your website. Besides there are ways to make the links draw less attention to them, still being underlined.
See this for example: https://css-tricks.com/styling-links-with-real-underlines/