  The Gowdy cosmologies provide a suitable arena to further develop Loop
Quantum Cosmology, allowing the presence of inhomogeneities. For the particular
case of Gowdy spacetimes with the spatial topology of a three-torus and a
content of linearly polarized gravitational waves, we detail a hybrid quantum
theory in which we combine a loop quantization of the degrees of freedom that
parametrize the subfamily of homogeneous solutions, which represent Bianchi I
spacetimes, and a Fock quantization of the inhomogeneities. Two different
theories are constructed and compared, corresponding to two different schemes
for the quantization of the Bianchi I model within the {\sl improved dynamics}
formalism of Loop Quantum Cosmology. One of these schemes has been recently put
forward by Ashtekar and Wilson-Ewing. We address several issues including the
quantum resolution of the cosmological singularity, the structure of the
superselection sectors in the quantum system, or the construction of the
Hilbert space of physical states.
