There are only three breeds of zebra in existence in the world. The Plains Zebra, the Mountain zebra and the Grevy's Zebra. There are a number of subspecies under the main species. If you check out this link
http://en.wikipedia.org/wiki/Zebra you can click on each species and subspecies. Both the mountain zebra and the Grevy's zebra have white under bellies. The Grevy's zebra has narrower stripes. Thought I'd share a little knowledge. Hope you enjoy.
Genus: Equus
Subgenus: Hippotigris
Plains Zebra, Equus quagga Quagga, Equus quagga quagga (extinct)
Burchell's Zebra, Equus quagga burchellii (includes Damara Zebra)
Grant's Zebra, Equus quagga boehmi
Selous' zebra, Equus quagga borensis
Chapman's Zebra, Equus quagga chapmani
Crawshay's Zebra, Equus quagga crawshayi
Mountain Zebra, Equus zebra Cape Mountain Zebra, Equus zebra zebra
Hartmann's Mountain Zebra, Equus zebra hartmannae
Subgenus: Dolichohippus
Grevy's Zebra, Equus grevyi