  Weyl semimetals are gapless quasi-topological materials with a set of
isolated nodal points forming their Fermi surface. They manifest their
quasi-topological character in a series of topological electromagnetic
responses including the anomalous Hall effect. Here we study the effect of
disorder on Weyl semimetals while monitoring both their nodal/semi-metallic and
topological properties through computations of the localization length and the
Hall conductivity. We examine three different lattice tight-binding models
which realize the Weyl semimetal in part of their phase diagram and look for
universal features that are common to all of the models, and interesting
distinguishing features of each model. We present detailed phase diagrams of
these models for large system sizes and we find that weak disorder preserves
the nodal points up to the diffusive limit, but does affect the Hall
conductivity. We show that the trend of the Hall conductivity is consistent
with an effective picture in which disorder causes the Weyl nodes move within
the Brillouin zone along a specific direction that depends deterministically on
the properties of the model and the neighboring phases to the Weyl semimetal
phase. We also uncover an unusual (non-quantized) anomalous Hall insulator
phase which can only exist in the presence of disorder.
