  We address the problem of finding conditions under which a compact Lorentzian
manifold is geodesically complete, a property, which always holds for compact
Riemannian manifolds. It is known that a compact Lorentzian manifold is
geodesically complete if it is homogeneous, or has constant curvature, or
admits a time-like conformal vector field. We consider certain Lorentzian
manifolds with Abelian holonomy, which are locally modelled by the so called
pp-waves, and which, in general, do not satisfy any of the above conditions.
%the condition that their curvature sends vectors that are orthogonal to the
vector field to a multiple of the vector field. We show that compact pp-waves
are universally covered by a vector space, determine the metric on the
universal cover, and prove that they are geodesically complete. Using this, we
show that every Ricci-flat compact pp-wave is a plane wave.
