  In this thesis we present the cosmological applications of some proposals
from Quantum Gravity. Namely, we will explore classical and quantum
cosmological implications of the Generalized Uncertainty Principle (GUP), the
Ho\v{r}ava-Lifshitz (HL) theory of gravity and the Swampland Conjectures.
Furthermore, we will also present a detailed analysis of Lorentzian Vacuum
Transitions in various contexts.
  First of all, we will present a study of the classical implications of a
general form of the GUP to the standard inflationary scenario driven by a
scalar field. We will then present the study of the compatibility between the
dS swampland conjecture and the Ho\v{r}ava-Lifshitz $F(\bar{R})$ theories. Then
we will study the effects of considering a GUP in the variables of the
superspace in a model of the Wheeler-DeWitt (WDW) equation in HL gravity.
Finally, we will present a general study of the Lorentzian vacuum transitions.
We will present a general method to compute the transition probabilities
between two minima of a scalar field potential for any model of the superspace
that leads to a generic form of the Hamiltonian constraint, by solving the WDW
equation with a semiclassical expansion. We will only consider up to first
order in this expansion but the method provides enough information to compute
the probabilities up to any desired order. We will then apply the method in
different scenarios.
  With these studies we will approach keys aspects of cosmology that are hoped
to be solved by quantum gravity, that is we will explore (at a toy model
level): the origin of inflation, the viability of the swampland conjectures and
a plausible model to avoid the initial singularity. This thesis encompasses six
research articles where these results were presented originally.
