If you are referring to the administration of the Catholic
Church, the date of its beginning is not certain . The final shape
of the administration of the Vatican, the curia, evolved over
time.
If you are referring to the curia as the senate house of the
ancient Romans, the first curia was built by the third king of
Rome, Tullus Hostilius (reigned 673 - 642 BC). It was called the
Curia Hostilia. The second curia was built by Lucius Cornelius
Sulla in 82 BC. It was called the Curia Cornelia. Julius Caesar
commissioned the construction of the third curia. However, he was
assassinated soon afterwards. Its construction was completed by
Augustus. It was called the Curia Julia.