Previously Kjarninn was a weekly digital news magazine served via Apple App Store aimed at tablet computer users but PDFs were also available at the website.

The founders of Kjarninn originally worked at 365's free newspaper Fréttablaðið, which maintained a virtual monopoly on the market (69% in 2004).

[1] In February 2013, Magnús Halldórsson, later a journalist at Kjarninn, criticised businessman and former owner Jón Ásgeir Jóhannesson, whose wife Ingibjörg Pálmadóttir was the owner of 365, for trying to influence the editorial independence of Fréttablaðið in his favour.

After the publication of the first issue the Financial Supervisory Authority demanded a report on Keflavik Savings Bank (by then defunct) that Kjarninn published be withdrawn.

[3] In October 2013 the Kaupthing winding up committee insisted Kjarninn withdraw the publication of a PriceWaterhouseCoopers report about how Kaupthing employees profited hugely by hedging against a strong Icelandic króna.