Küng represented Switzerland at the World Championships in 2013 in Schladming, Austria. After placing 18th in the Super G, he produced his best run of the 2013 season in the downhill, finishing in 7th position.
Küng's first World Cup win came in Super G in December 2013, at Beaver Creek, USA. His second came a month later at the downhill in Wengen, Switzerland. Through January 2015, he has two World Cup wins and five podiums.