There are only four verses in the New Testament that speak explicitly about the "catching up", or the "rapture", or the "gathering of the elect". All four are very straightforward. They’re extremely easy to understand.

"Let no one deceive you in any way. For that day will not come, unless the falling away comes first, and the man of lawlessness is revealed…"  2 Thes 2:3

Immediately after the tribulation of those days the sun will be darkened, and the moon will not give its light, and the stars will fall from heaven, and the powers of the heavens will be shaken. Then will appear in heaven the sign of the Son of Man, and then all the tribes of the earth will mourn, and they will see the Son of Man coming on the clouds of heaven with power and great glory. And he will send out his angels with a loud trumpet call, and they will gather his elect from the four winds, from one end of heaven to the other" Matt 24:29-31

"Behold! I tell you a mystery. We shall not all sleep, but we shall all be changed, in a moment, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, and the dead will be raised imperishable, and we shall be changed."  1 Cor 15:51-52

"For the Lord himself will descend from heaven with a cry of command, with the voice of an archangel, and with the sound of the trumpet of God. And the dead in Christ will rise first. Then we who are alive, who are left,will be caught up together with them in the clouds to meet the Lord in the air, and so we will always be with the Lord." 1 Thes 4:16-17

This last trumpet is the final of seven trumpets listed in the Book of Revelation (Rev 11:15-18).

If we put these passages together we learn that this "gathering" or "catching up" occurs: when the Lord “comes”, “after the tribulation”, after the great falling away and the revealing of the "man of lawlessness", and at the last trumpet.