  This survey article reviews recent results on fermion system in discrete
space-time and corresponding systems in Minkowski space. After a basic
introduction to the discrete setting, we explain a mechanism of spontaneous
symmetry breaking which leads to the emergence of a discrete causal structure.
As methods to study the transition between discrete space-time and Minkowski
space, we describe a lattice model for a static and isotropic space-time,
outline the analysis of regularization tails of vacuum Dirac sea
configurations, and introduce a Lorentz invariant action for the masses of the
Dirac seas. We mention the method of the continuum limit, which allows to
analyze interacting systems. Open problems are discussed.
