Is there such thing as eye transplant? I heard about it, but how technologically advanced are we?

No. Not an entire eye transplant yet. Corneas are often transplanted. When an eye is removed, an ocular prosthesis is placed. We are advanced enough to put retinal chips into eyes though.
No. We cannot transplant the entire eye currently. We are able to transplant a cornea from a donor eye to a recipient with a diseased or cloudy cornea.