Born in Fiesch in the canton of Valais, Albrecht made his World Cup debut at age 19 in January 2003 in a slalom at Schladming, Austria.

Eight months later he won his first World Cup race, a super combined held in Beaver Creek, Colorado, followed by a giant slalom victory three days later.

[1] At Kitzbühel, Austria, on Thursday, 22 January 2009, Albrecht crashed in the final downhill training run on the Hahnenkamm's Streif course; he sustained brain and lung trauma and was placed in an induced coma.

[5][6][7][8] Albrecht expectedly missed the remainder of the 2009 season but had recovered sufficiently to train with the Swiss ski team in October, less than nine months after the crash.

During a training run for the Lake Louise downhill in November 2012, Albrecht crashed and suffered a dislocated left kneecap with torn ligaments, and underwent surgery in Switzerland.