One of the more interesting things I've come across is a open source airplane group called MakerPlane which lists their vision as:
“We want more people to safely and quickly build and fly their own airplanes.”The emphasis on MakerPlane is an airplane that can be built with small tools, like using a CNC router to cut foam or plastic pieces and laying them up with fiberglass. From my vantage point, it seems the majority of homebuilt airplanes are made of fiberglass and foam (like these kits). Behind that are riveted aluminum planes, and there are certainly some made of wood and fabric. I have different friends that have built each kind.
Basically we are designing an aircraft that can be built on a computer controlled mill at home, or at a makerspace which is easy to assemble and quick to build. The plans and instructions will be available for free to anyone that wants them!
There's just one big problem. It's a hobby project and the design isn't done yet. This is a rendering of the tentative design.
Experimental Aircraft Association, the EAA. See their Hints for Homebuilders video page and the general homebuilding channel. There are chapters of the EAA around the country, and a lot of homebuilders out there. The EAA also runs classes and workshops around the country to teach folks the techniques required to build their dream plane. There's a fire hose of information out there. Enjoy!
Don't forget there are helicopters out there, too. (H/T to Kerodin)