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Dispensationalism is a Christian theology holding that time is divided into seven different "social contracts" (for lack of a better term) that God has with the world. We are currently in the "Dispensation of Grace" according to this belief system. In the history of Christianity, it's practically brand spanking new, being invented in the 1830s by a fundamentalist Christian sect in Scotland called the Plymouth Brethren.
The pre-tribulation rapture belief comes from Dispensationalism and was not known in any part of Christendom before the Plymouth Brethren (or possibly the Irvingites - see immediately below) invented it.
Some scholars have traced the pretribulation-rapture belief to a "divine revelation" received by a member of an 1830s proto-Pentecostal sect in Scotland, the Catholic Apostolic Church (a/k/a the Irvingites), which belief was picked up by the Plymouth Brethren's John Darby, tacked onto his emerging Dispensationalism theory, and claimed as his own.
 Reference material
So where did today's evangelicals come up with this stuff? The notion traces back to John Darby and was then adopted by Plymouth Brethren member Cyrus Scofield in his Scofield Reference Bible, a version of the King James Bible with extensive commentary (and theological fabrication) from Scofield. It spread from there to American evangelicalism's culture as a whole.
The word "rapture" itself to refer to Jesus grabbing all the living Christians off the earth did not appear until the 1909 publication of the Scofield Reference Bible. The word "rapture" is not found anywhere in the text of the Bible itself.
The Scofield Reference Bible is still popular today.
 In pop culture
You know all that Left Behind stuff? The song "I Wish We'd All Been Ready"? The A Thief in the Night movie series? Jack Chick's end times paranoia? Hal Lindsey's bestseller The Late Great Planet Earth? The theology they teach at Bob Jones University? The "WARNING: In Case Of Rapture Somebody Grab The Wheel" bumper stickers? All of those are based on belief in a "pretribulation rapture".
 Typical Dispensationalist behaviors
Dispensationalists are notorious for trying to set the date of Christ's return, tying Bible prophecy to current events, tying Bible prophecy to politics involving the state of Israel, and naming the Antichrist. In the 1830s the original Dispensationalists in the Plymouth Brethren and Irvingites were naming Robert Owen (American founder of several anarchist-socialist communes such as New Harmony, Indiana) as the likely Antichrist. William Branham predicted the World Council of Churches and the Catholic Church would merge by 1977 and the rapture would be in 1977. Hal Lindsey thinks the founding of modern Israel in 1948 set off the prophetic clock which Jesus must return within a generation after, predicted Jesus' return by 1988 and wrote The 1980s: Countdown to Armageddon. Mary Stewart Relfe has the same view of Israel as Lindsey, and named Anwar Sadat as the Antichrist. Jack Chick has the same view of Israel, but thinks the Pope is the Antichrist. Pat Robertson (whom some of us think is the Antichrist) seemingly wants to start World War III as soon as possible to speed up Jesus' return. Harold Camping set the date of Christ's return in 1994, and when that didn't happen, started preaching that we have left the "Dispensation of Grace" and nobody can now be saved. All of them think the final sequence of events leading to the rapture and Christ's return will be set off by an invasion of Israel by Russia.
Some evangelists know better than to openly try to set the date of Christ's return or name the Antichrist, so they do it in fiction form instead. For example, the Left Behind books and films by Tim LaHaye, and the A Thief in the Night films by Russell Doughten. That way when events don't unfold exactly as they said, they can say it was "just fiction".
 Non-Dispensationalist Evangelicals
Unlikely as it seems, there are also some evangelicals and fundamentalists who do not subscribe to Dispensationalism, nor to its related doctrines like the pre-millennial, pre-tribulation rapture. You have 30 seconds to think of one.
I bet you can't because they have been so drowned out by the Left Behind and Late Great Planet Earth mania that most people assume Dispensationalist theology is evangelical Christian theology.
The alternative viewpoints include Praeterism, which is the belief that the Bible prophecies in the books of Book of Daniel and Book of Revelation were already fulfilled in the first or second century; Postmillennialism, which is the belief the world will get better and better as the Christian faith spreads and that Jesus will return after a 1000 year era of peace and prosperity; the post-tribulation believers who believe Christians will remain on the earth during the 7 year reign of the Antichrist just like everyone else (sorry, you don't get to be raptured out of the coming hard times and you are all going to be "left behind"); and amillennialism, the view that the Bible prophecies were not literal to begin with.
The Orthodox Presbyterian Church, a fundamentalist Reformed body, does not permit dispensationalists to be ordained ministers. The Lutheran Church - Missouri Synod, a Confessional Lutheran body that is officially creationist, holds to amillennialism.
 Bible passages supporting the Dispensationalist viewpoint
 Scientific research supporting the Dispensationalist viewpoint
Many people thought President G.W. Bush's "No Child Left Behind" initiative was a federally-funded program to ensure all students meet certain educational standards before they are allowed to advance. In reality, it was about making sure all the children accepted Jesus Christ as their Personal Lord and Savior so they would be caught up into the air during the imminent Rapture.
 See also
Some other concepts also not found anywhere in the Bible, which have nonetheless become core beliefs of today's evangelical Christianity: