  This is a complete and exhaustive review on the so-called holographic axion
model -- a bottom-up holographic system characterized by the presence of a set
of shift symmetric scalar bulk fields whose profiles are taken to be linear in
the spatial coordinates. This simple model implements the breaking of
translational invariance of the dual field theory by retaining the homogeneity
of the background geometry and therefore allowing for controllable and fast
computations. The usages of this model are very vast and they are a proof of
the spectacular versatility of the framework. In this review, we touch upon all
the up-to-date aspects of this model from its connection with massive gravity
and effective field theories, to its role in modeling momentum dissipation and
elastic properties ending with all the phenomenological features and its
hydrodynamic description. In summary, this is a complete guide to one of the
most used model in Applied Holography and a must-read for any researcher
entering this field.
