Each time a cylinder's compressed charge is ignited a violent force is applied to the top of the piston, through the connecting rod, and into the crankshaft. The flywheel absorbs some of that violence and stores the energy in the form of motion (inertia). As some have noted, a single cylinder 4-stroke engine might be totally incapable of running without a flywheel because it could not keep rotating long enough to get back to the power stroke.
I would say that most other engines could run without a flywheel, but the engine - and everything attached to it - would experience considerable vibration. The more cylinders, the less vibration. But adding cylinders adds complexity and, importantly, friction, so there is a practical limit to the number of cylinders an engine can have.
The other end of the spectrum is the "hit-and-miss" engine, in which a single cylinder fires only occasionally and a ginormous flywheel keeps things moving in the interim. Cool beasties, and you can probably see one at your local "heritage days" festival.
http://en.wikipedia.org/wiki/Hit-and-miss_engine