Current practice
The 1983 reform of the Roman Catholic Church's Code of Canon Law has streamlined the procedure considerably compared to the process carried out previously. (See below.)
The process begins at the diocesan level, with the bishop giving permission to open an investigation of the virtues of the person who is suspected of having been a saint. This investigation may not open until permission is given by the Vatican, and not sooner than five years after the death of the person being investigated. However, the pope has the authority to waive this waiting period, as was done for Mother Teresa. When sufficient information has been gathered, the subject of the investigation is called "Servant of God", and the process is transferred to the Roman Curia—the Congregation for the Causes of the Saints—where it is assigned a postulator, whose task is to gather all information about the life of the Servant of God. When enough information has been gathered, the congregation will recommend to the pope that he make a proclamation of the Servant of God's heroic virtue, which entitles him or her to receive the title "Venerable". A Venerable has as of yet no feast day, but prayer cards may be printed to encourage the faithful to pray for a miracle wrought by his or her intercession.
The next step depends on whether the Venerable is a martyr. For a martyr, the pope has only to make a declaration of martyrdom, which then allows beatification, yielding the title "Blessed" and a feast day in the Blessed's home diocese and perhaps some other local calenders. If the Venerable was not a martyr, it must be proven that a miracle has taken place by his or her intercession. Today, these miracles are almost always miraculous cures, as these are the easiest to objectively establish. (The patient was sick, there was no known cure for the ailment, prayers were directed to the Venerable, the patient was cured, and doctors cannot explain it.)
To pass from Blessed to Saint, one (more) miracle is necessary. The declaration of sainthood is considered to be infallible, while declarations of venerability and beatitude are not. A saint's feast day is considered universal, and may be celebrated anywhere within the Catholic church, although it may not appear on the general calendar.
In the case of persons that common usage has called saints from "time immemorial" (in practice, since before 1500 or so), the Church may carry out a "confirmation of cultus", which is much simpler. For example, Saint Hermann Joseph had his veneration confirmed by Pope John Paul II.
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