I must have clicked something naughty in Compiz Config Settings Manager, because I lost the top and left side panels, and generally there was not much more than just the desktop background. I was stumped on how to fix this.
Then I found:
http://www.tuxgarage.com/2011/04/missing-top-and-side-panels-in-unity.html
Very good walkthrough for fixing Unity.
In the end, I executed the "Reset Compiz" section, and things are back to normal.
