Zombies can be stabbed or taken down by being slaughtered, ignited by fire or having their heads exploded.
There are several ways to prevent the dead from resurrecting, although none are as effective as bursting their brains. This is a random critical hit, much like it was in Resident Evil 2 Remake, and the key to triggering it is to shoot zombies squared. I'll say it again: it's hit and miss, and pumping lead into their skulls doesn't guarantee a meaty, Scanners-style explosion. That said, a zombie with its brain outside is a permanently dead zombie (Lickers aside, naturally).
If you take down a zombie with its brain firmly intact, you can still remove it from combat. Using your knife or combat weapon (assuming you have the spare ammo), kill or shoot the zed until it coils up with a dying gasp. Remember that if save he's about to attack, so back out. By the way, the knife doesn't break in Resident Evil 3, so if you're not surrounded by zombies, this is a great way to conserve ammo and keep the bastards down.
Finally, the explosive barrels and fire will kill the zombies, although burning them to death might take a while (and they'll keep attacking you until they drop). Use the environment to your advantage, as an explosive cannon at the right time can kill a herd of zombies with a single bullet.
It's worth mentioning that some zombies will return no matter what you do as part of a scripted event. You can clean up a room, pump round after round through every rotten front you see, and see the dead resurrected on the way back. This is not common, but if a zombie appears dead when you enter an area and a bullet does not wake it up, expect it to attack you later.
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