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The first registered domain name ever was Symbolics.com, by Cambridge, Massachusetts computer company Symbolics Inc. In addition to being the oldest domain name, it's also the longest running and you can still visit it today, though it's no longer under operation by its original owners, who sold the domain in 2009 after 25 years of ownership.
Today Symbolics.com is owned by a small investor group in Irvine, CA, which bought the domain for an undisclosed amount, presumably as a means of marketing its services, though there is a certainly a cool factor in owning the web's first domain name.
Symbolics Inc. was a spinoff from the MIT AI Lab, who engineered and manufactured a line of Lisp machines (designed to run software coded with the Lisp programming language), which were the first commercially available general-purpose workstations.
The second domain name to be registered was bbn.com (BBN Technologies) followed by think.com (Thinking Machines) in April and May of 1985. Other more recognizable names at the time followed, the likes of DEC (Digital Equipment Corporation), which was bought by Compaq/HP later on, registered dec.com in September 1985. Xerox registered xerox.com in January 1986, and became the seventh domain name. HP, IBM, Sun and Intel all registered their domain names in March 1986.
* The only domain names that existed before belonged to the public sector and were limited to organizations with access to the U.S. Department of Defense's Advanced Research Projects Agency (ARPA).