  Cosmic voids are promising cosmological laboratories for studying the dark
energy phenomenon and alternative gravity theories. They are receiving special
attention nowadays in view of the new generation of galaxy spectroscopic
surveys, which are covering an unprecedented volume and redshift range. There
are two primary statistics in void studies: (i) the void size function, which
characterises the abundance of voids, and (ii) the void-galaxy
cross-correlation function, which contains information about the density and
velocity fields in these regions. However, it is necessary a complete
description of the effects of geometrical (Alcock-Paczynski effect, AP) and
dynamical (Kaiser effect, RSD) distortions around voids in order to design
reliable cosmological tests based on these statistics. Observational
measurements show prominent anisotropic patterns that lead to biased
cosmological constraints if they are not properly modelled. This thesis
addresses this problematic by presenting a theoretical and statistical
framework based on dynamical and cosmological foundations capable of describing
all the underlying effects involved: the expansion effect (t-RSD), the
off-centring effect (v-RSD), the AP-volume effect and the ellipticity effect
(e-RSD). These effects can be understood by studying the mapping of voids
between real and redshift space. In this way, we lay the foundations for a
proper modelling of the aforementioned statistics. In addition, we present a
new cosmological test based on two perpendicular projections of the correlation
function. The method is fiducial-cosmology free, which allows us to effectively
break any possible degeneracy between the cosmological parameters involved.
Moreover, it allows us to significantly reduce the number of mock catalogues
needed to estimate covariances.
