  We present a review on basic aspects of quantum cosmology in the presence of
cut-off physics. We first analyze how the Wheeler-DeWitt equation in a pure
metric approach describes the quantum Universe, showing how the singularity is
not removed. We then discuss basic features of loop quantum cosmology. For the
isotropic Universe, we compare the original $\mu_0$ scheme and the most
commonly accepted formulation, the $\bar{\mu}$ scheme. Some results concerning
the Bianchi Universes are also discussed. Finally, we consider some relevant
criticisms on the real link between full loop quantum gravity and its
minisuperspace implementation. In the second part of the review we consider the
isotropic Universe and the Bianchi models in the framework of polymer quantum
mechanics. We first address the polymerization in terms of the
Ashtekar-Barbero-Immirzi connection and show how the resulting dynamics is
isomorphic to the $\mu_0$. Then we analyze the polymerization of volume-like
variables, and we see the resulting dynamics is isomorphic to the $\bar{\mu}$
scheme with a fixed critical energy density. Finally, we consider the polymer
quantum dynamics of the homogeneous and inhomogeneous Mixmaster model through a
metric approach. We compare the results obtained by using the volume variable
to the standard Misner variable. In the latter case we find a cosmology that is
still singular, and its chaotic properties depend on the ratio between the
lattice steps for the isotropic and anisotropic variables. We conclude the
review with some considerations on changing variables in the polymer
representation. In particular we consider how the dynamics can be mapped in two
different sets of variables (at the price of using a coordinate-dependent
lattice step), and we infer some possible implications on the equivalence of
the $\mu_0$ and $\bar{\mu}$ scheme of loop quantum cosmology.
