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At the time Jesus lived, demon possession was prevalent and demonic spirits entered into and possessed both humans and animals. However, this phenomenon was temporary and localized to the time of Jesus’ ministry on earth, solely in order to demonstrate His authority over the powers and forces of darkness and evil (Satan and his demons).
The modern superstition and belief in demon possession and the practice of “exorcism” today is not Scriptural. The Roman Catholic Church and Pentecostal (charismatics) churches in particular are known for this type of sensationalism and superstitious practices; but it is simply yet one more of their many false doctrines/false teachings which come from paganism, not the true Christianity of the New Testament Bible.
Wayne Jackson writes: 
Do evil spirits enter into human bodies and afflict people today? I confidently affirm they do not. Unfortunately, though, some modern writers have argued that demon activity is still a part of Earth’s environment.
Charles Ryrie contended that certain “fallen angels” are “still free to roam the earth as demons carrying out Satan’s designs” (1959, 296).  Merrill Unger, a respected scholar, subtitled his book, Biblical Demonology, “A Study of the Spiritual Forces Behind the Present World Unrest.” 
And several years ago a book titled UFOs, Satan and Evolution enjoyed a limited circulation in the evangelical community. Therein the author claimed that hundreds of UFO visits to Earth represented an invasion of demons. He cited one “example” where a demon raped a woman—an interesting feat for a spirit! The fact that a prominent creationist wrote the foreword for this literary fiasco remains an inexplicable mystery.
The position that demon possession does not exist today can be argued from a twofold base. First, a thoughtful study of the details associated with the so-called modern examples of demon habitation reveals that these cases bear no resemblance to the genuine examples of spirit possession described in the New Testament. The contrast is dramatic.
Second, a consideration of certain data set forth in the New Testament leads only to the conclusion that demon possession was a first-century experience; it was allowed for a very specific reason, and the divine concession was suspended near the end of the apostolic era.
 Ryrie, Charles C. 1959. Biblical Theology of the New Testament. Chicago, IL: Moody.  Unger, Merrill F. 1952. Biblical Demonology. Wheaton, IL: Scripture Press.