  Loop Quantum Gravity is a nonperturbative and background independent program
for the quantization of General Relativity. Its underlying formalism has been
applied successfully to the study of cosmological spacetimes, both to test the
principles and techniques of the theory and to discuss its physical
consequences. These applications have opened a new area of research known as
Loop Quantum Cosmology. The hybrid approach addresses the quantization of
cosmological systems that include fields. This proposal combines the
description of a finite number of degrees of freedom using Loop Quantum
Cosmology, typically corresponding to a homogeneous background, and a Fock
quantization of the field content of the model. In this review we first present
a summary of the foundations of homogeneous Loop Quantum Cosmology and we then
revisit the hybrid quantization approach, applying it to the study of Gowdy
spacetimes with linearly polarized gravitational waves on toroidal spatial
sections, and to the analysis of cosmological perturbations in preinflationary
and inflationary stages of the Universe. The main challenge is to extract
predictions about quantum geometry effects that eventually might be confronted
with cosmological observations. This is the first extensive review of the
hybrid approach in the literature on Loop Quantum Cosmology.
