Muslims believe in the rapture.
Breeze from the South (The "Rapture")
“The Hour will not come until Allah takes away the best people on earth; only the worst of people will be left; they will not know any good or forbid any evil.”
Narrated Uqbah son of Amir:
It has been narrated on the authority of AbdurRahman son of Shamasah al-Mahri who said: I was in the company of Maslamah son of Mukhallad, and Abdullah son of Amr son of 'As was with him. Abdullah said: "The Hour shall come only when the worst type of people are left on the earth. They will be worse than the people of pre-Islamic days. They will receive whatever they ask of Allah."
While we were still sitting, Uqbah son of Amir came and Maslamah said to him: "Uqbah, listen to what Abdullah says." Uqbah said: "He knows better so far as I am concerned." I heard the Messenger of Allah (peace be upon him) say: "A group of people from my Ummah will continue to fight in obedience to the Command of Allah, remaining dominant over their enemies. Those who oppose them shall not do them any harm. They will remain in this condition until the Hour overtakes them."
(At this) Abdullah said: "Yes. Then Allah will raise a wind, which will be fragrant like musk and whose touch will be like the touch of silk; (but) it will cause the death of all (faithful) persons, not leaving behind a single person with an iota of faith in his heart. Then only the worst of men will remain to be overwhelmed by the Hour."
Abdullah son of 'Amr said,
“The Prophet (S) said, 'The Dajjal will appear in my Ummah, and will remain for forty - "I cannot say whether he meant forty days, forty months or forty years." - Then Allah will send ‘Eyssa son of Maryam, who will resemble 'Urwah son of Masud. He will chase the Dajjal and kill him. Then the people will live for seven years during which there will be no enmity between any two persons. Then Allah will send a cold wind from the direction of Syria, which will take the soul of everyone who has the slightest speck of good or faith in his heart. Even if one of you were to enter the heart of a mountain, the wind would reach him there and take his soul. Only the most wicked people will be left; they will be as careless as birds, with the characteristics of beasts, and will have no concern for right and wrong. Satan will come to them in the form of man and will say, "Don't you respond?" They Will say, "What do you order us to do?" He will order them to worship idols, and in spite of that they will have sustenance in abundance, and lead comfortable lives…’”
Muslim version of 'The Rapture'
Note: In Christian Prophecy, there is something similar to this, that is referred to as the "Rapture", which is almost identical to what the "cold wind from Syria" does, except that Christians believe the rapture to occur upon the believers to save them from the tribulation of the antichrist before the return of "Jesus", while Muslim s believe the rapture to occur upon the believers to save them from the destruction of the world on the Day of Judgment after the return of "Jesus".
Narrated by Aisha:
I heard Allah's Apostle (peace be upon him) said: "The (system) of night and day would not end until the people have taken to the worship of Lat and Uzza."
I said: Allah's Apostle (peace be upon him) I think when Allah has revealed this verse: "He it is Who has sent His Messenger with right guidance, and true religion, so that He may cause it to prevail upon all religions, though the polytheists are averse (to it)" (Qur’an 9:33), it implies that (this promise) is going to be fulfilled.
Thereupon he said: "It would happen as Allah would like. Then Allah would send the sweet fragrant air by which everyone who has even a mustard grain of faith in Him would die and those only would survive who would have no goodness in them. And they would revert to the religion of their forefathers. "
Hafiz Ibn-i-Hajar has alluded to a tradition from the Prophet, saying:
"The Doomsday shall not come until things come to such a pass that a woman shall pass through a crowd of men and one of the men will rise and lift her undergarment (as a tail of a she-lamb is lifted) and engage in copulation with her. Seeing such a happening one of the men will say: "It would have been better if you had engaged in intercourse with her in privacy behind a wall." "Then", said the Prophet, "such a man will be as virtuous among those people as Abu Bakr and 'Umar are the most righteous among you."
(Tabrani in Fath-ul-Bari)
Note: A similar incident of "copulation in public" has in fact occurred.
Narrated Abu Hurayrah: Allah's Apostle said,
"The Hour will not be established till the buttocks of the women of the tribe of Daus move while going round Dhi-al-Khalasa."
Note: Dhi-al-Khalasa was the idol of the Daus tribe, whi