The promise of 72 virgins to Islamic martyrs is the most notorious of these rumors. The simple answer is there is no such promise in the Quran.
The section that the controversy arises from is Chapter 78, misinterpretation of verse 33.
I won't insult our Muslim friends here by poorly transcribing Quranic texts, so I will summarize.
The verses (29-34) speak of a success, or fulfillment of all desires awaiting the righteous, including gardens, vineyards, and a full cup, but differ in the description of companions, but virginity is not specified. Nor is any number given.
Not to be too far down the rabbithole- but Arabic is a chore to translate, other sources are often necessary to embellish Quran Arabic sufficiently into another language. This creates variance. Variants of religions are nothing new.
