Zombies are real, to an extent.
There are three types of 'zombies'. The Haiti voodoo zombie, the undead zombie and the viral zombie.
The word zombie comes from the Kimbundu word 'zumbi' which means one without spirit, in complete basics.
Voodoo zombies are people poisoned by a cocktail of extremely toxic extracts which put the victim into a comatic state. The people is thought dead and buried. During this state the brain becomes severly damaged due to the poisoning and lack of oxygen from being buried. Later, the voodoo priest would dig up the zombified person who will have no memory of his former life and only be able to understand basic functions. A slave, and a zombie, if you will.
Now, actual corpses rising after death to feast on living flesh? No, that's not possible. When a body ceases to function, it's brain and organs shut down, that's it. The body becomes a rock, it's a shell, it's absolutely impossible to animate the body. So no, undead zombies are nada.
Finally, although it has never happened, i wont ever rule out the real possibility of a viral outbreak such as the one seen in 28 Days Later. In that movie they are not actually zombies of course, just people infected with a virus which takes away everything from them except basic instint and of course, deep rage. I don't see why such a virus couldn't actually exist, something which could be transfered through saliva or blood and make the host violent, etc.
Undead, no.
Voodoo, yes.
Viral, plausable.