This article shares information about registering a domain at the university.
If you want a domain name ending with .org, .net, etc., you will need to use an outside domain name provider. Use a hyphen to separate multiple words within the domain name. You cannot use a space or an underscore. It is preferable to register domains outside arizona.edu and just use your own DNS servers---e.g., OpenDNS, or the nameservers provided by your registrar of choice.
You can also configure the domain name to use the UA's name servers:
Once completed, submit your domain information using the DNS Request form.