The overwhelming majority of mainstream Christian denominations reject the notion of reincarnation and consider the theory to challenge basic tenets of their beliefs. Certain churches indirectly address the subject through teachings about death.
A few consider the matter open to individual interpretation due to the few biblical references which survived the purging of texts considered to be heretical in the founding years of Christianity as a church. Some Christians contend that reincarnation was taught by the early Christian church, but due to bias and mistranslations, these teachings were lost or obscured.
Many of the philosophies associated with the theory of reincarnation focus on "working" or "learning" through various lifetimes to achieve some sort of higher understanding or state of "goodness" before salvation is granted or acquired.
Basic to Traditional Christianity is the doctrine that humans can never achieve the perfection God requires and the only salvation is total and complete forgiveness accomplished through the sacrifice Jesus made on the cross wherein he took the sins of mankind. There seems to be evidence however that some of the earliest Christian sects such as the Sethians and followers of the Gnostic Church of Valentinus believed in reincarnation, and they were persecuted by the Romans for this.
A number of Evangelical and (in the USA) Fundamentalist Christian maintain that any phenomena suggestive of it are deceptions of the devil. Although the Bible never mentions the word reincarnation, there are several passages through New Testament that reject reincarnation or the possibility of any return or contact with this world for the souls in Heaven or Hell (see Hebrews 9:27 and Luke 16:20-31)
The Bible contains passages in the New Testament that could be taken to allude to reincarnation. In Matthew 11:10-14 and 17:10-13, John 1:21, the Jews ask John the Baptist if he is Elijah and John replies clearly that he is not, implying that Jesus' reference was meant in a figurative sense (which is what most Christians accept). It should be noted that Elijah never actually "died," but was "raptured" in a chariot of fire.
Furthermore, the prophetic texts stated that God would send Elijah back to Earth, as a harbinger of Jesus Christ. As cousins they were born respectively to barren Elizabeth and Zacharias; Jesus, firstborn of Mary and Joseph, was the first to rise from the dead visibly demonstrating his power over death. It can also be taken to mean an apparition, not a reincarnation.
There are various contemporary attempts to entwine Christianity and reincarnation. Geddes Macgregor, wrote a book called Reincarnation in Christianity: A New Vision of Rebirth in Christian Thought, Rudolf Steiner wrote Christianity as Mystical Fact and Tommaso Palamidessi wrote Memory of Past Lives and Its Technique which contains several methods which are supposed to help in obtaining memories from previous lives.
Several groups which consider themselves to be Christian and support reincarnation include the Christian Community, the Liberal Catholic Church, Unity Church, The Christian Spiritualist Movement, the Rosicrucian Fellowship and Lectorium Rosicrucianum. The Medieval sect known variously as the Cathars or Albigensians who flourished in the Languedoc believed in Reincarnation, seeing each soul as a fallen angel born again and again into the world of Matter created by Lucibel (Lucifer).