What is an airworthiness certificate?
An airworthiness certificate is an FAA document which grants authorization to operate an aircraft in flight.

Who may apply for an airworthiness certificate?
A registered owner or owner's agent of an aircraft may apply for an airworthiness certificate.

Are there different classifications of airworthiness certificates?
Yes.  There are two different classifications of FAA airworthiness certificates: Standard Airworthiness Certificate, and Special Airworthiness Certificate.

Who is authorized to issue an FAA airworthiness certificate?
Only FAA Aviation Safety Inspectors and authorized Representatives of the Administrator (i.e.Designees), as defined in 14 CFR Part 183, "Representatives of the Administrator", are authorized to issue an FAA airworthiness certificate.

Can the FAA revoke an airworthiness certificate?
Yes.  The FAA can revoke an existing airworthiness certificate in any category (14 CFR section 21.181), if the aircraft no longer meets its approved design and/or is not in an airworthy condition.

Does the FAA provide information regarding the definition of the term "airworthy"?
The FAA provides information regarding the definition of the term "airworthy" in FAA Order 8130.2, Airworthiness Certification of Aircraft and Related Products, Chapter 1.