A mathematician from Morgan Freeman's show Through The Wormhole (which sadly isn't available here, legally at least) shows how fast an hypothetical zombie virus could spread around the world. The epidemic would be inevitable and cities would be overran in weeks. If you don't feel that's worrying wait until you hear it from Freeman's epic voice.
I'm not a fan of zombies and I'm quite fed up with them already. But the possibility of a yet unknown lethal virus spreading that fast seems very real to me. And that's truly scary.