Zombies are melee units that wander the wastes. They cannot pillage your Improvements; they’re just not interested in that. But they will go immediately for any unit wandering about outside, so make sure that you keep an eye open when leaving your cities. They can also attack your cities, turning any destroyed Population into more zombies...
When a unit dies on the map, it carries a slight chance of returning as a Zombie unit some turns later. However, if a unit is destroyed by a Zombie, it will certainly rise again on the following turn. The undead menace steadily grows, though – for each Zombie killed anywhere in the world, remaining and new Zombies get stronger and stronger. Make sure you know what you’re getting into when you go to war with the dead!
Holy Sites and Campuses hold a temporary solution to the zombie horde. Using the Turn Undead project at a Holy Site, or the Dark Signal project at a Campus will allow you to briefly take control of Zombies within your city limits, meaning that you can turn the horde against your enemies. Just be sure to be safely out of reach when your control expires… or when you strategically end it. In addition, your Spies can spread the deadly curse to your enemies with the Zombie Outbreak action.