The practice of haruspicy, the name for this kind of divination, was said to have originated among the ancient Etruscans. A bronzesculpture of a liver, complete with the name of regions marked on it assigned to various gods, has been found at Piacenza, and has been connected to the practice of haruspicy.
The art of haruspicy was taught in the Libri Tagetici, a collection of texts attributed to Tages, a childlike being who figures in Etruscan mythology, and who was discovered in an open field by Tarchon. Haruspicy continued to be practiced throughout the history of the Roman empire; the emperor Claudius was a student of Etruscan and opened a college to preserve and improve their art, which lasted until the reign of Theodosius I. In 408, the haruspices offered their services when the Goths under Alaric threatened Rome; Pope Innocent I reluctantly agreed to allow them to help so long as the rituals were kept secret.
External link:
The Art of Haruspicy (http://www.cs.utk.edu/~mclennan/BA/Har.html) by John Opsopaus. Contains a complete how-to.
They were drawn from members of the ruling class of Rome, and were organised in 'colleges' and sub-groups with particular functions.
For example, there were pontifices (pontiffs), augurs (associated with interpretation of auspices - signs given by the gods through the flight of birds, thunder, lightning, and other natural phenomena), haruspices (originally of Etruscan origin, consulted about prodigies), flamines or individual gods, and fetiales, associated with the declaration of war.
The chief priest was known as the pontifex maximus, a title that was subsequently used by Roman Catholic popes.