It has been an open question for some time whether for every noncommutative division ring D finite dimensional over its center, the multiplicative group D^* = D \ {0} has a maximal proper subgroup. We prove that the answer is yes, except possibly for some unusual division algebras that probably don't exist. The difficult case in the proof is that of a quaternion division algebra over a Euclidean field, which we handle by a geometric argument. This is joint work with R. Hazrat.