There is an obvious ethnic subdivision within Jews, between the Sephardic and the Ashkenazi. In several vaguely antisemitic websites (see for instance, here, here, and here ), but also in a completely un-anti-Semitic book The Thirteenth Tribe: The Khazar Empire and Its Heritage, (Arthur Koestler 1976), there is this claim that the Ashkenazi Jews are largely descended from the Khazars, who converted to Judaism in the 7-8th century. The medieval conversion of the Khazars to Judaism is well accepted, and among other places, it is described in Yehudah Ha-Levi's famous 12th century classic, The Kuzari.
There is some political baggage floating with this idea, since it tends to get interpreted as denying modern Jews a Biblically derived right to settle in Israel. This makes it difficult to get a straight answer from googling around. I would prefer to avoid politics, and I also would prefer to avoid anti-Semitism, just to settle the geneological question with unbiased research. I think that modern genetic tools should be able to easily trace the ancestors of Ashkenazi Jews, and determine the percentage of admixture of the central Asian, European, and Semitic elements, and tell us exactly what their genetic history was.
So to what extent are Ashkenazi Jews descended from Khazars? Can it be determined, or has the interbreeding diluted the genetic markers beyond any hope of reconstruction of those long ago migrations?
Just to disclose my biases: my father is Sephardi, my mother Ashkenazi, and I don't think any person has any claim to any land other than a deed.