If you've ever ridden a bicycle, you may have noticed that, past a certain initial velocity, the bike pretty much does stays upright on its own. Sure, you're there to make major changes (like steering, stopping, and increasing speed), but a properly built bike is able to coast on its own and maintain balance. You can test this by pushing your rider-less bike at a reasonable rate of speed and watching it "drive itself" until it loses speed and falls over.

Okay, so why does this happen? What does physics tell us about how the bike works? Three minutes of explanation from MinutePhysics explains. Yes, there's some math in hereā€”but I caught most of it simply by watching the (repeated) demos and observing how the bike stopped steering itself when it was slightly modified. Behold: