A desalination and power plant was also partially destroyed in an attack, according to the government.
No deaths have been reported in connection with the incidents. Kuwait's air defenses tried to stop missiles and drones aimed at targets in the country overnight.
The refinery is one of the largest in the Middle East.
Israel reports that Iran also fired a number of missiles at the country on Friday. There were initially no reports of casualties.
In the United Arab Emirates, operations at a gas plant were halted due to a fire that broke out after air defenses thwarted an attack.
