Katrín Jakobsdóttir has ruled the country for the past four years.
She is the most popular candidate for the post of Prime Minister, but her Green Left has lost support.
Today, the three governing parties have a narrow majority of 33 seats.
At the same time, several parties appear to be passing the five percent barrier.
As many as nine parties may take seats in the Althingi, which could open up new coalitions.
At nine o'clock local time, the polling stations opened and then closes at 10 pm, ie at midnight Swedish time.