The Association for Unmanned Vehicle Systems International has (and has had for many years) a number of student competitions. There are so many competitions, so many years of competitions, so many teams, and so many details for each team (sponsors, designs, papers, etc) that I can only give the briefest of overviews here.
AUVSI International Aerial Robotics Competition - Apparently this competition has been held since 1991. The Joint Architecture for Unmanned Systems has defined a communications protocol for unmanned vehicles, and offers $750 for any team that can demonstrate basic proficiency with this message set. The JAUS has specifications that can be used in the Aerial Robotics Competition, and also in a couple of the other competitions.
The main competition has $80,000 in prizes, and involves an airborne robot that can fly 3km to find a town, identify a building and any openings, and get pictures of a target inside the building. The pictures have to get back to the beginning location. As you can see, the competitions have a military flavor. The competition is held at Ft. Benning.
There are links to the registered teams at the competition site. Many, but not all, of the designs are helicopter-based.
2008 UAS Competition - This is an Unmanned Aerial System competition with a similar-sounding mission to the above one. However, this scenario is in support of the U.S. Marine Corps. There will be a pre-defined mission, route, and objective to photograph, but as circumstances change, new objectives or routes may need to be taken. A quick mission is important. "At least $20,000 in prizes have been awarded the past two years".
Intelligent Ground Vehicle Competition - This competition is for a rugged unmanned ground vehicle that can carry a 20 pound payload. It has to be able to navigate an obstacle course. It brings to mind the DARPA Grand Challenges. There are numerous monetary awards for this competition.
International Autonomous Underwater Vehicle Competition - This is brought by the AUVSI and ONR (Office of Naval Research). It's for unmanned submarines (AUVs). The challenge involves control (depth, speed, direction), accousting and visual sensing, and dropping or getting an object. There's a wealth of information about this year's teams (each has a website and paper) here.
International Autonomous Surface Vehicle Competition - This is also brought by the AUVSI and ONR. It's for unmanned boats. They need to be able to navigate, avoid obstacles, and find GPS-based objectives marked by a buoy.
These are all for Earth-based applications, but they seem relevant to me to many space applications, too. I picture satellites, Europa submarines, Mars or Lunar rovers, planetary air vehicles, human-assisting robots floating in a space station, and so on when I read these competitions and think about the space "angle". In fact, I would not be surprised if there are great opportunities with these competitions for space organizations to participate or interact with this unmanned vehicle community.