When and where were horseshoes invented?

We...actually aren't sure. In fact, what people call "hoof boots" predates the nailed on horseshoe, being used by the Romans.

We know that horseshoes appeared in northern Europe just after the Roman period, so they may have been invented in wet, muddy climates where hoof boots, especially primitive ones, would tend to be pulled off of the horse's feet. But we still don't know who first nailed a crescent of iron onto a horse's hoof (or what the horse thought about it!).