Scotland's supreme criminal court remains the High Court of Justiciary.
The swearing-in saw Lord Phillips of Worth Matravers become the first President of the Court.
He was joined by 10 other colleagues in taking an oath of allegiance to uphold the law. A final 12th member of the court will be appointed at a later date.
Lord Phillips said: "This is the last step in the separation of powers in this country. We have come to it fairly gently and gradually, but we have come to the point where the judges are completely separated from the legislature and executive.
"The change is one of transparency. It's going to be very much easier for the public to come to our hearings.
"I would hope that the court is still sitting in 100 years' time and that when people look back at this step that they see it as a very significant step in the constitution of this country."
While only the Law Lords' judgements were televised from Parliament, all of the Supreme Court's hearings will be open to the public.
Its building, in Parliament Square, Westminster, includes a public cafe and education facilities. For the first time, television cameras have been fitted into the courtrooms meaning that many hearings will be available to broadcasters.
Supreme Court: Independent of Parliament
The Supreme Court sits for the first time later on Thursday to deal with a relatively minor issue relating to legal costs.
Its first major appeal hearing follows next week in a case concerning terrorist suspects whose assets have been frozen.
Although the actual business and workings of the justices will be essentially the same as those of the Law Lords, it will be watched closely to see if the move across Parliament Square will affect the way its decides cases.
One group of influential solicitors and barristers is launching a blog to monitor the Supreme Court's decision-making.
But others have criticised the change, arguing that it is largely a cosmetic exercise.
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