Another approach, in addition to carrying stuff to fix a flat, is to use tires that have some inherent properties towards preventing flats. Kevlar belts come to mind. And maintaining enough air pressure to minimize the possibility of pinch flats.
Those of us who are superstitious believe that the act of carrying a pump and flat repair supplies in itself lowers the odds of suffering a flat.

I carry a Mini Morph on my single bikes and a Mountain Morph on our tandem.