The discovery of the top quark and the measurement of its mass were mportant milestones in experimental physics. Given the much larger data-sets expected in Run II, due to begin in 2001 at Fermilab, it should be possible, in principle, to substantially reduce the uncertainty in our knowledge of the top quark mass. However, this will require better methods of mass measurement as well as improvements in event simulation. Guided by our experience from Run I, we have initiated a program of work to develop neural network based methods of mass measurement that could be useful during Run II. We report the results of some preliminary studies of top quark mass measurements in the electron-muon channel.