  We study in detail a variety of gravitational toy models for
hyperscaling-violating Lifshitz (hvLif) space-times. These space-times have
been recently explored as holographic dual models for condensed matter systems.
We start by considering a model of gravity coupled to a massive vector field
and a dilaton with a potential. This model supports the full class of hvLif
space-times and special attention is given to the particular values of the
scaling exponents appearing in certain non-Fermi liquids. We study linearized
perturbations in this model, and consider probe fields whose interactions mimic
those of the perturbations. The resulting equations of motion for the probe
fields are invariant under the Lifshitz scaling. We derive
Breitenlohner-Freedman-type bounds for these new probe fields. For the cases of
interest the hvLif space-times have curvature invariants that blow up in the
UV. We study the problem of constructing models in which the hvLif space-time
can have an AdS or Lifshitz UV completion. We also analyze reductions of
Schroedinger space-times and reductions of waves on extremal (intersecting)
branes, accompanied by transverse space reductions, that are solutions to
supergravity-like theories, exploring the allowed parameter range of the hvLif
scaling exponents.
