Loebner Prize: http://www.loebner.net/Prizef/loebner-prize.html
Chatterbox Challenge: http://www.chatterboxchallenge.com/
AI-Challenge: http://aichallenge.org/
Darpa challenges: http://www.darpa.mil/
Health Prize: http://www.heritagehealthprize.com/c/hhp
Patent challenge: http://community.topcoder.com/ntl/?page_id=536
Data Competitions: http://www.kaggle.com/