Very low-mass stars, brown dwarfs, and giant planets with effective temperatures < 2700 K (or spectral types cooler than M7) are usually referred to as ultra-cool dwarfs. Their low atmospheric temperatures and high pressures favor the appearance of condensates, which may draw together to form cloud-like structures in their atmospheres. According to state-of-the-art models, these dusty clouds are composed by particles from refractory elements (e.g., Ti, Fe, Cr, etc) for late-M and L dwarfs, and sulfide particles for T dwarfs.

In this thesis work, we studied the optical  (RIZ) and near-infrared (JHKs) linear polarimetry of young (1-500 Myr) and mature (<500 Myr) ultra-cool dwarfs spanning the spectral types M7-T2. The main objective was to demonstrate the presence of dust in ultra-cool atmospheres of different surface gravities and rotation periods. We also studied the linear polarimetric signal of our own planet, as if it were an exo-Earth, to guide the analysis of future discoveries of Earth twins around other stars. For the optical wavelengths, we exploited the imaging and spectroscopic capabilities of the "Andalucía Faint Object Spectrograph and Camera" (ALFOSC) on the Nordic Optical telescope (NOT), and "the FOcal Reducer and low dispersion Spectrograph" (FORS) on the Very Large telescope (VLT), and for the near-infrared we used the "Long-slit Intermediate Resolution Infrared Spectrograph" (LIRIS) on the William Herschel telescope (WHT).

In our first study, we investigated the predicted relation between linear polarization at near-infrared wavelengths (1.03 and 1.25 micron) and oblate shapes due to fast rotation for 18 ultra-cool dwarfs with projected rotation velocities (vsini) <30 km/s. We found that ~40 % of the sample showed linear polarization in the range 0.4-0.8 %. Additionally, our data hint at ultracool dwarfs with the highest rotation (vsini< 60 km/s) having a factor of about 1.5-2 larger in the J-band linear polarimetry detection fraction than ultra-cool dwarfs with lower vsini. However, these results contain the degeneracy of the rotation axis inclination. Thus, in a second work we monitored a sample of 18 ultra-cool dwarfs with vsini>30 km/s and thought to be older than 0.5 Gyr (i.e. they have a likely radius of ~1 Rjup according to evolutionary models) in the I-band by using the Liverpool and the IAC80 telescopes. We derived the rotation periods of 9 targets (7 of them for the first time) and the most likely range of inclination angles for their rotation axes. When considering values of linear polarization detected with a confidence level of 3-sigma, we found a hint of correlation between large values of linear polarization and short rotation periods. We also found that objects with rotation periods <2 h tend to exhibit more dispersion in their linear polarimetric values taken on different occasions.

In a third study, we examined the likely dependence of optical linear polarimetry (R- and I-bands) with the rotation phase for the M8.5 dwarf TVLM 513-46546, which was known to show variability in its intensity light curve and had a single measurement of J-band linear polarization. We found that this dwarf shows a periodic variability of ~1.98 h with an amplitude of 0.5 % in its intensity light curve and a degree of linear polarization that changes from <0.2 % up to ~1.3 % in the I-band, which shows that the linear polarimetric properties of ultra-cool dwarfs cannot be infered from individual single-epoch measurements.

In a fourth study, we investigated the dependency of linear polarization with age at optical and near-infrared wavelengths for a sample of low- (1-500 Myr) and high-gravity (>500 Myr) ultra-cool dwarfs. We found similar values of linear polarization for both samples of ultra-cool dwarfs, no dependence of linear polarization with spectral type, and polarimetric variability with peak to peak amplitudes up to 1.5 % on scales of about a rotation, in two out of four targets that were monitored over several hours. Long-term polarimetric variability is also detected in nearly all dwarfs of the sample with data spanning months to years.

In the last chapter, we analyzed the main molecular features present in the polarimetric spectra of the Earthshine at optical (0.4-0.9 micron) and near-infrared wavelengths (0.9-2.4 micron). We found the highest values of linear polarization (>10 %) at the bluest wavelengths and a decrease towards redder wavelengths until reaching a nearly flat value (~4.5 %) beyond ~0.8 micron. We also reported several molecular features due to O2 and H2O that show linear polarimetry degrees above that of the continuum. These features will become a powerful tool for characterizing Earth-like planets in polarized light.