Can I attach a camera to the outside of my aircraft in the USA?
The NTSB actually recommended the use of exterior cameras to the FAA in 2012.

It appears to be something of a grey area, but it looks like you should contact your local FSDO for approval of an external load if you're permanently fixing a camera to a plane.

The grey area comes in that if the camera is temporarily attached, you shouldn't need special approval, but in practicality it could come loose and damage the plane.

GoPro says this about affixing their cameras to planes:


  Industrial strength suction cup. Strong enough to pull dents out of a car door or stay attached to an airplane at 200mph+, proven though not endorsed by GoPro per FAA rules.


There's a forum thread where someone asked the FSDO to sign it off as a Minor Alteration, and it was determined that a short logbook entry would suffice:


  Temporarily installed a GoPro camera mounted to a standard inspection panel on (Left/Right) lower wing in place of the standard inspection panel. There are no electrical connections associated with this installation. This is a minor alteration as it has no appreciable effect on performance, flight characteristics, or other qualities affecting airworthiness (refer to FAR 1.1 Major Alteration and FAR 21.93). A net weight change of 9 oz is incurred with the temporary installation; therefore, the weight and balance change is negligible (refer to AC 43.13 1B, 10-2 (c)).