Apostolic Succession

In Christianity, the doctrine of Apostolic Succession (or the belief that the Church is 'apostolic') maintains that the Christian Church today is the spiritual successor to the original body of believers in Christ composed of the Apostles. Different Christian denominations interpret this doctrine in different ways.
In episcopal churches, the Apostolic Succession is understood to be the basis of the authority of bishops (the episcopate). Specifically in the case of the Roman Catholic Church, the Apostolic Succession as passed on through Saint Peter is also the basis for the specific claim of papal primacy. Within the Anglican Communion this is seen more as a symbolic precedence, not unlike the Eastern Orthodox Patriarch of Constantinople. In any event, all these communions recognize Apostolic Succession as the determining criterion of a particular group's legitimacy as a Christian Church.

Mainstream Christianity

Apostolicity as episcopal continuity
The Catholic Church (including its Oriental and Eastern rites), Eastern Orthodox, Oriental Orthodox, Assyrian, Anglican, Old Catholic, Independent Anglican, Independent Catholic, and some Lutheran Churches hold that apostolic succession is maintained through the consecration of their bishops in unbroken personal succession back to the apostles but these Churches do not necessarily interpret this "succession" identically. In Catholic and Orthodox theology, the unbrokenness of apostolic succession is significant because of Jesus Christ's promise that the "gates of hell" (Matthew 16:18) would not prevail against the Church, and his promise that he himself would be with the apostles to "the end of the age" (Matthew 28:20). According to this interpretation, a complete disruption or end of such apostolic succession would mean that these promises were not kept as would an apostolic succession which, while formally intact, completely abandoned the teachings of the Apostles and their immediate successors; as, for example, if all the bishops of the world agreed to abrogate the Nicene Creed or to repudiate the Bible.
Both Orthodox and Catholics believe that each of their teachings today is the same as or is in essential harmony with the teaching of the first apostles, although each might deny this about the other, at least where the teachings of each are in conflict. This form of the doctrine was formulated by Irenaeus of Lyons in the second century, in response to certain Gnostics. These Gnostics claimed that Christ or the Apostles passed on some teachings secretly, or that there were some secret apostles, and that they (the Gnostics) were passing on these otherwise secret teachings. Irenaeus responded that the identity of the original Apostles was well known, as was the main content of their teaching and the identity of the apostles' successors. Therefore, anyone teaching something contrary to what was known to be apostolic teaching was not, in any sense, a successor to the Apostles or to Christ.
Catholics recognize the validity of the apostolic successions of the Eastern Orthodox, Oriental Orthodox, Assyrian, Old Catholic, and some Independent Catholic Churches (such recognition is not reciprocated by the Eastern Orthodox, who do not separate "valid" from "licit"). Pope Leo XIII clarified, in his 1896 bull Apostolicae Curae that the Catholic church believes that the Anglican Church's consecrations are invalid because of changes made to the rite of consecration under Edward VI, thus denying that Anglicans participate in the apostolic succession; the Church of Sweden's apostolic succession is seen as having been maintained, and following the establishment of the Porvoo Communion an increasing number of Anglicans will also be able to trace their succession through Swedish bishops as well as Old Catholic bishops, whose holy orders are recognized as valid by Rome and who, at least those of the Union of Utrecht, are in full communion with Canterbury since the Bonn Agreement of 1931. It should also be noted that since the issuance of Apostolicae Curae, many Anglican jurisdictions have revised their ordinals, bringing them more in line with ordinals emanating from the early Church.
In addition to a line of historic transmission, Eastern Orthodox and Oriental Orthodox churches additionally require that a hierarch maintain Orthodox Church doctrine, which they hold to be that of the Apostles, as well as communion with other Orthodox bishops. The Eastern Orthodox have permitted clergy ordained by Catholic and Anglican bishops to be rapidly ordained within Orthodoxy. However, this is a matter of ekonomia and not recognition of Apostolic Succession, although in some cases, Catholic priests entering Eastern Orthodoxy have been received by "vesting" and have been allowed to function immediately within Orthodoxy as priests.
The Armenian Apostolic Church, which is one of the Oriental Orthodox churches, recognizes Catholic episcopal consecrations without qualification (and that recognition is reciprocated).
Some Protestant churches, such as Anglicans (including those known in the U.S. as Episcopalians), the Church of Sweden, the Evangelical Lutheran Church of Finland, and the Evangelical Lutheran Church of Latvija, do have Apostolic Succession (also known as the "historic episcopate"). Bishops in the United Methodist Church do not claim to be within the historic episcopate. Their succession derives from John Wesley who was an ordained priest of the Church of England, but not himself a bishop and therefore had no power to consecrate others. He justified his practice of ordaining "elders" ("presbyters") for Methodism by appealing to a perceived need and by citing a minority opinion among the early Church Fathers (and possible ancient precedent from the Church of Alexandria) which held that presbyters ("priests" or "elders") could, at least collectively, indeed ordain other such presbyters and even consecrate, or "set apart" bishops.

Apostolicity as doctrinal continuity
Most Protestant churches would deny that the apostolicity of the Church rests on an unbroken episcopacy. They generally hold that one important qualification of the apostles was that they were chosen directly by Jesus and that they witnessed the resurrected Christ. According to this understanding, the work of these twelve (and the Apostle Paul), together with the prophets of the twelve tribes of Israel, provide the doctrinal foundation for the whole church of subsequent history through the Scriptures of the Bible. To share with the apostles the same faith, to believe their word as found in the Scriptures, to receive the same Holy Spirit, is the only sense in which apostolic succession is meaningful, because it is in this sense only that men have fellowship with God in the truth (an extension of the Reformation doctrines of sola fide and sola scriptura). The most meaningful apostolic succession for most Protestants, then, is the faithful succession of apostolic teaching.
It is worth noting, however, that some Protestant charismatic churches include "apostles" among the offices that should be evident into modern times in a true church, though they never trace an historical line of succession.
Those who hold to the importance of episcopal apostolic succession would counter the above by appealing to the New Testament, which, they say, implies a personal apostolic succession (from Paul to Timothy and Titus, for example) and which states that Jesus gave the Apostles a "blank check" to lead the Church as they saw fit under the guidance of the Holy Spirit (Matthew 18:18 and Acts Chapter 15, for example). They appeal as well to other documents of the very early Church, especially the Epistle of St. Clement to the Church at Corinth, written around 96 CE. In it, Clement defends the authority and prerogatives of a group of "elders" or "bishops" in the Corinthian Church which had, apparently, been deposed and replaced by the congregation on its own initiative. In this context, Clement explicitly states that the apostles both appointed bishops as successors and had directed that these bishops should in turn appoint their own successors; given this, such leaders of the Church were not to be removed without cause and not in this way. Further, proponents of the necessity of the personal apostolic succession of bishops within the Church point to the universal practice of the undivided early Church (up to 431 CE), from which, as organizations, the Roman Catholic, Eastern Orthodox, Oriental Orthodox, the Assyrian, and the Churches of the Anglican Communion are all directly descended.
At the same time, no defender of the personal apostolic succession of bishops would deny the importance of doctrinal continuity in the Church. As stated above, Ireneus explicitly ties the two together.

Latter-day Saints (Mormons)
The Church of Jesus Christ of Latter-day Saints (LDS) has a similar, but unique position. Latter-day Saints believe that God directs his church at all times through revelation in response to prayer. However, individuals are entitled to revelation only for that calling over which they have authority. For example, everyone is entitled to revelation concerning themselves; a head of household is entitled to revelation for his family; a bishop has the authority to receive revelation concerning the congregation over which he presides (e.g. Rome or Ephesus). Only ordained apostles have the authority from the Lord to receive revelation for doctrine for the entire church. An excellent example of church-wide apostolic revelation is Acts 10:1-48 where Peter had prayed and received revelation from God that the gospel could now go forward to the Gentiles as well as the Jews.
Hence, Latter-day Saints have a different interpretation of the scripture where Christ says "upon this rock I will build my church":
"When Jesus came into the coasts of Caesarea Philippi, he asked his disciples, saying, Whom do men say that I the Son of man am? And they said, Some say that thou art John the Baptist: some, Elias; and others, Jeremias, or one of the prophets. He saith unto them, But whom say ye that I am? And Simon Peter answered and said, Thou art the Christ, the Son of the living God. And Jesus answered and said unto him, Blessed art thou, Simon Bar-jona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven. And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it."
Matthew 16:13-18
When Christ asks his disciples who they think he is, Peter had the right answer because he prayed and received revelation: "Blessed art thou, Simon Bar-jona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven." When Christ says upon this rock I will build my church", the rock of which he was speaking was revelation. As long as there were apostles with the authority from the Lord to receive revelation for the entire church, the church would continue to follow Christ. Circumstances change and so must the church: at one time Christ said not to preach to the Gentiles, and later Peter was given revelation when it was time to start.[1] The need for constant ongoing revelation is critical to conduct the affairs of the church.
The LDS Church understands that Christ chose apostles and gave them the authority to receive revelation for the church by the laying on of hands. As necessary, the apostles passed this authority to others by choosing and ordaining new apostles by the laying on of hands (such as Paul and Matthias). Those individuals then had the appropriate authority to receive revelation for and officiate over the church in that office at that time:
"And they prayed, and said, Thou, Lord, which knowest the hearts of all men, shew whether of these two thou hast chosen, That he may take part of this ministry and apostleship, from which Judas by transgression fell, that he might go to his own place. And they gave forth their lots; and the lot fell upon Matthias; and he was numbered with the eleven apostles."
Acts 1:24-26
However, due to distance and persecution, the existing apostles were not able to meet frequently enough to choose and ordain new apostles, and eventually all of the apostles were killed, leaving nobody in the church with apostolic authority.
At this point, for the church to continue properly, apostolic authority would have needed to be dispensed again from God, but this did not happen. Instead, the bishops took over the leadership of the church, assuming authority that they did not possess - breaking the aspostolic succession. An example of the resulting confusion is the First Council of Nicaea where Roman Emperor Constantine the Great, a Sun god worshipper, presided over a meeting of bishops to determine church doctrine, instead of that doctrine being revealed by God to worthy apostles. The LDS church refers to the resultant loss of revelation and falling away from the teachings of Jesus Christ as the Great Apostasy. Latter-day Saints believe that this was predicted when Amos said that there would be a "famine of hearing the words of the Lord" in Amos 8:11.
The LDS Church maintains that the authority from God needed to be restored to the earth, which took place when God the Father and His son, Jesus Christ, appeared to Joseph Smith, Jr. near Palmyra, New York in 1820 and called Joseph as a prophet to restore Christ's church to the earth with correct doctrines and practices.
Near the time that Joseph formally organized the church in 1830, the apostles Peter, James and John appeared to Joseph. They laid hands on Joseph and restored to him the apostolic authority to govern the church.[2] Joseph was visited by other heavenly messengers at different times, each one conferring upon him the particular authority or keys for which they had stewardship. John the Baptist restored the Aaronic Priesthood, Peter James and John restored the Melchizedek Priesthood, with other heavenly messengers such as Moses and Elijah restoring additional authority. Joseph was given the authority like the apostles of old, to confer to others specific priesthood authority by the laying on of hands. All of the various keys of this authority have been and are passed on to worthy, male members of the LDS Church according to their particular offices. In this way, Latter-day Saints hold that apostolic authority was restored to the earth through the original twelve apostles and apostolic succession continues today through the ordination of new apostles as the older apostles pass away.

Jehovah's Witnesses
Jehovah's Witnesses teach that apostolic succession is an erroneous doctrine. They base this teaching on several lines of reasoning. Four of which are:
1. Peter was not the "rock" the 'church' was built on. [1]
2. There is no evidence Peter was in Rome. [2]
3. The claimed line of successors from Peter to the Pope is questionable. [3]
4. They contrast the replacement of Judas Iscariot and the death of the apostle James. The account names James as successor to Judas. Acts does not mention any successor being named for the faithful apostle James. [4]