Kaplansky's conjectures

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if $K$ is a field and $G$ is a torsion-free group, then the group ring $K[G]$ has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and present my recent counterexample to the unit conjecture.