Abstract: This will be a brief review of the zero-divisor conjecture due to Kaplansky: the product of two nonzero elements in the group ring is always nonzero. (It is false as I state it here, and I would challenge the audience to find a counterexample before coming to the talk.) Surprisingly, among other things this conjecture is related to questions about the Murray-von Neumann dimension of certain Hilbert modules. The talk will provide more questions than answers.