The chart below shows that the Quran they have today is actually a 1936 revision of the 1924 Cairo Edition of the Quran, which comes from one of 4 recorded variations of the Hafs Quranic tradition, which was derived from one of two variations of the Asim Quranic tradition, which itself was one of 3 variations of the Kufi version of the Uthmanic Quranic tradition which, for its part, was just one of many Quranic collections from DECADES after the death of Muhammad.
In short, today's Quran is "a revision of a revision of a variant of a variant of a version of a version of the original Quran," whatever it might have looked like. Bart Ehrman would have a field day with the Quran if he were ever to find the guts to examine its textual history.
Let's reverse that whole sequence to better assist us in understanding how Muslims got their modern Quran.
First there was the "original Quran."
Then Muslims produced a version from that original.
From that first version came another version.
That second version had a variant.
That variant had a variant.
Then that variant was revised.
The revision was also revised.
And that final version is the Quran most Muslims have today.
Remember that the actual compilation of the Quran only happened after Muhammad died. He was long dead when all this revising and choosing was being done, so Muslims have no way of knowing if those doing the selecting got the right one since "the Last Prophet" was dead and nobody could ask him to approve it. Muslims will never know if their "non-prophets" chose the correct one.
The above pic is from http://research-islam.blogspot.com. Much more information is available there as well.
Updating this post...
In a recent debate on this topic which is linked to below, Australian seminary professor Dr. Bernie Power presented dozens of different Arabic Qurans which should settle the issue once and for all. Contrary to the often heard Muslim claim, Islam has never had only one Quran.
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Has the Quran been changed? A Muslim-Christian Debate