There are several verses in the New Testament that allude to a rapture or "taking away" of the Christian church.
1 Thessalonians 4:16-17 16For the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first:
17Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord.
1 Corinthians 15:51-52 51Behold, I show you a mystery; We shall not all sleep, but we shall all be changed,
52In a moment, in the twinkling of an eye, at the last trump: for the trumpet shall sound, and the dead shall be raised incorruptible, and we shall be changed.
Jesus gave two different descriptions for his coming for believers.
THe first is in Matthew 24:29-31. It says " 'the sun will be darkened,
and the moon will not give its light;
the stars will fall from the sky,
and the heavenly bodies will be shaken.'
"At that time the sign of the Son of Man will appear in the sky, and all the nations of the earth will mourn. They will see the Son of Man coming on the clouds of the sky, with power and great glory. And he will send his angels with a loud trumpet call, and they will gather his elect from the four winds, from one end of the heavens to the other.
This describes Christ coming for his believers, whom he calls the 'elect'.
Also this passage talks about Jesus coming for the believers prior to a time of great tribulation here on the earth.
Matthew 24:37-42 - As it was in the days of Noah, so it will be at the coming of the Son of Man. For in the days before the flood, people were eating and drinking, marrying and giving in marriage, up to the day Noah entered the ark; and they knew nothing about what would happen until the flood came and took them all away. That is how it will be at the coming of the Son of Man. Two men will be in the field; one will be taken and the other left. Two women will be grinding with a hand mill; one will be taken and the other left. Therefore keep watch, because you do not know on what day your Lord will come.
Jesus really did talk alot about the end of the world and coming for those who put their trust in Him. In the book of John he tells his followers that he will go and prepare a place for them, (believers) then come again a get them and go to a place he has prepared for them in heaven.
So to say there really isn't much in the Bible about it is untrue imo.
Its a fascinating subject imo and yes one that even believers of yester-year believed in. (maybe not established organized churches, but certainly individual believers.
I found out there are a lot of varying opinions on this after googling the word "rapture"