After discovering them in Peter Frankopan's "Silk Roads", I was fascinated by the history of the Khazars, a Turkic steppe empire which held judaism as their state religion. However, I was even more intrigued by the fact that once their empire fell, their culture seems to have "vanished".
I have recently stumbled upon the hypothesis that Ashkenazi communities were actually descended from the Khazars. However, the wikipedia article very quickly shoots down this theory, claiming it has been used by antisemitic or anti-zionist groups. One does indeed stumble upon very... weird... blogs when googling the word "Khazar".
However, the hypothesis seems to be defended by some respected if controverted authors, like Shlomo Sand.
What is the general historical consensus on the Khazar hypothesis, and if they indeed have no relation with the Ashkenazis, then what happened to them?
While you're waiting for an answer that could be tailored to your questions, you might find these threads useful:
Some of these are quite old so keep that in mind
This AMA by u/John_LathamSprinkle went into religion in the Khazar Khaganate, and the fact that the population followed a mix of the religions that they were surronded by as well as Judaism.
These three comments focus on the question of descent, by u/gingerkid1234
Third one, much more recent than the other two
This answer by a deleted user that is also quite (maybe a bit more than quite) old, goes into who the Khazars were.