A semi-Riemannian manifold is geodesically complete (or for short, complete) if its maximal geodesics are defined for all times. For Riemannian metrics the compactness of the manifold implies completeness. In contrast, there Lorentzian metrics on the torus that are not complete. Nevertheless, completeness plays an important role for fundamental geometric questions in Lorentzian geometry such as the classification of compact Lorentzian symmetric spaces and in particular for a Lorentzian version of Bieberbach's theorem. We will study the completeness for compact manifolds that arise from the classification of Lorentzian holonomy groups, which we will briefly review in the talk. These manifolds have abelian holonomy and carry a parallel null vector field. By determining their universal cover we show that they are complete. In the talk we will explain this result and further work in progress, both being joint work with A. Schliebner (Humboldt-University Berlin).