Flury grew up in Davos Monstein and learned to ski at the age of three. As an eleven year old she competed in the first interregional and national youth races. In December 2008 she took part in FIS races for the first time, and she achieved her first victory at this level in February 2009. Her first appearances in the European Cup were in January 2010; at the end of the same year she drove into the points for the first time. It was rather surprising that Flury won the Swiss Super G championship title in March 2011. She made her first podium in the European Cup on February 7, 2012 in Jasná . She made her World Cup debut on January 11, 2014 in Zauchensee , where she finished in 33rd place in the downhill. With a total of five participations in Junior World Championships , two fourth places were her best results (2013 and 2014 in the downhill).
In the 2014/15 season, Flury won three European Cup races. This made her third in the overall standings as well as second in the downhill and super-G classifications. This meant a fixed starting place for the following World Cup season. But because of a hip injury, she was unable to contest a single race in the 2015/16 winter. She won her first World Cup points on December 18, 2016 with 11th place in the Super-G in Val-d'Isère. On December 9, 2017, Flury celebrated her first World Cup victory in the Super-G in St. Moritz , which was also her first podium finish in the World Cup.
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