Zigbee, not zeebee. (Zeebee is something else, but it is not used by smartthings.)
Unlike Zwave, Zigbee allows for multiple different profiles, so a hub might support Zigbee (like Control4) But still not be able to use the devices that work with smartthings. See the FAQ for more details (the topic title is a clickable link)
SmartThings supports two Zigbee profiles: ZHA 1.2 and parts of the newest profile, Zigbee 3.0. So again, some devices will work with some other hubs, but not all.
The following post lists Hubs currently on the market which support both Zwave and the zigbee profiles that smartthings supports. There aren’t many, but there are some. 
Alternative Hubs - #186 by JDRoberts
Homeseer does not support Zigbee. It is possible with home assistant, but not as easy as zwave.
Also remember that you will probably need some Zigbee Mains-powered devices to act as repeaters. (Z wave repeats only for Z wave, zigbee repeats only for Zigbee.)
The IKEA Tradfri plug-in pocket sockets make good Zigbee repeaters and cost around $10 each when purchased directly from IKEA.