Apostolic Succession: what is it, whose got it and who wants it?
Apostolic Succession is the belief that the Christian Church today is the spiritual successor to the original body of believers in Christ composed of the Apostles. In the Catholic Church this succession is understood to be carried through the sacrament of Holy Orders through the episcopal ordinaries, the bishops. The Pope is the successor of Saint Peter particularly. The Apostolic Succession of the Orthodox Church is recognized by Rome, which is one of the reasons that their sacraments are recognized as valid.
Now to the muddied waters. The Church holds that all bishops have the inherent ability to ordain a baptized male to be a deacon, priest, or bishop. A valid but illicit ordination, as the name suggests, is one where a bishop uses his valid ability to ordain someone whom under canon law or instruction from the pope he was prohibited from ordaining, it therefore being illicit.
However such an ordination is only valid if the properly performed. In the case of ordination this is the laying of hands and the proper charge to priesthood. The Anglican Church, for example is not generally seen to have Apostolic Succession because during the reign of Edward VI the outward forms of ordination in the Anglican Church were changed. Pope Leo XIII wrote a Papal Bull Apostolicae Curae stating that fact. Amazingly some modern Catholic theologians have question this. Further muddying the waters is the fact that in 1922 the Ecumenical Patriarch in Constantinople affirmed that the Orthodox Church does recognize the validity of the Anglican Church's Apostolic Succession and of its Holy Orders. Of course this was before the Anglican Church started ordaining women. Pope Benedict has reaffirmed the Anglican Holy Orders to be “absolutely null and utterly void.”
Another group claiming Apostolic Succession is the Old Catholic Church. The Old Catholic Church stakes its claim on succession though the Apostolic Vicar of Utrecht, the Dutch Archbishop, Cornelius van Steenoven, who had a falling out with Rome. After Vatican I, which the Old Catholic Church refused to recognize, the Church spread to America and other parts of the world. The ordinations of Archbishop van Steenoven are seen by the Roman Catholic Church as valid but illicit.
Can a bishop created by a valid but illicit ordination in his turn perform ordinations? The answer is yes...but the person ordained must be someone who can otherwise by lawfully ordained. For example an unbaptised person could not be validly ordained, nor can a woman. A married man can be validly ordained, and indeed there are married priest in both the Orthodox and Roman Catholic Churches.
Unfortunately for the Old Catholics, like the Anglicans, they have taken to ordaining women, which would invalidate any succession in those lines since a woman "bishop" is not a true bishop and has no ability to ordain anyone.
Here's where it get complicated. Many Anglo-Catholics and traditionalist Old Catholics have doubts about the directions of their respective Churches, and care about Apostolic Succession. These members of their respective clergies have personally sought out bishops who, by their lights, might still have true Apostolic Succession via an unbroken line of valid (but illicit) ordinations back to a known Roman Catholic or Orthodox bishop to lay hands upon them and ordain them. Odd isn't it? These break away Christians look to Rome (and our Orthodox brothers) to secure the validity of their priestly orders?
What about Protestants in general? Most don't care about Apostolic Succession at all. I won't even get into the non-Christian Churches, such as the Latter Day Saints.