It hasn't yet been established that they ARE zombies. They behave like 'zombies' from certain movies, but bullets seem to stop them. Are they rotting? Are they just rabid humans? Do chest wounds injure them? Are their hearts beating, do they digest food or starve to death? We don't know, NONE of this has been answered yet. We can't know what caused it without first knowing what the 'zombies' ARE.

Who/what are the 'usual suspects'?

It could be a virus, a fungus, an infectious Prion (Look up 'Prions' if you wanna read something REALLY scary; they're infectious proteins that spread exponentially and can't be stopped and they kill you fast), mass-mutations, or nanobots- and that's only things from KNOWN science. What if this plague comes from some totally undiscovered branch of science? That'd be like the Black Death back in the day- people were baffled about what was causing it. We know what it was now (Bubonic plague) because we have microscopes etc. What if the modern world doesn't have the tools yet to even DETECT whatever's causing the 'Zombie Plague'?

My crazy theory; What if some weird radiation pattern from a Black Hole (or a Gamma Ray burst) caused instantaneous mutations in billions of humans? Something like a Gamma Ray burst would hit the entire planet all at once, even the side facing away from it; maybe only certain sets of genes (or proteins) are affected, found ONLY in humans and maybe chimps, so MOST humans are mutated in a matter of hours and the rest only get the 'plague' via a bite. Other animals and plants wouldn't be affected cause they don't share the target chromosome (or protein)

Can it be cured? No. But once the Zombies are all wiped out the survivors can repopulate the world.