The elusive neutrinos are among the most intriguing constituents of the particle zoo. The observation of neutrino flavour oscillations, indicating that neutrinos are massive, provides the only direct evidence for physics beyond the Standard Model. Neutrinos imprint peculiar signatures in the Cosmic Microwave Background (CMB) and in the distribution of Large-Scale Structure (LSS) in the Universe, making cosmology a very promising arena for probing neutrino properties. A detection of neutrino masses is avowedly among the key goals of several upcoming CMB and LSS surveys. For such a promise to be robustly realized, a number of issues need to be addressed, particularly on the LSS side. In this thesis, I describe a number of recent important developments in neutrino cosmology on three fronts. Firstly, focusing on LSS data, I will show that current cosmological probes (and particularly galaxy power spectrum data) contain a wealth of information on the sum of the neutrino masses. I will report on the analysis leading to the currently best upper limit on the sum of the neutrino masses of 0.12 eV. I show how cosmological data exhibits a weak preference for the normal neutrino mass ordering because of parameter space volume effects, and propose a simple method to quantify this preference. Secondly, I will discuss how galaxy bias represents a severe limitation towards fully capitalizing on the neutrino information hidden in LSS data. I propose a method for calibrating the scale-dependent galaxy bias using CMB lensinggalaxy cross-correlations. Another crucial issue in this direction is represented by how the bias is defined in first place. In the presence of massive neutrinos, the usual definition of bias becomes inadequate, as it leads to a scale-dependence on large scales which has never been accounted for. I show that failure to define the bias appropriately will be a problem for future LSS surveys, leading to incorrectly estimated cosmological parameters. In doing so, I propose a simple recipe to account for the effect of massive neutrinos on galaxy bias. Finally, I take on a different angle and discuss implications of correlations between neutrino parameters and other cosmological parameters. I show how, in non-phantom dynamical dark energy models (which include quintessence), the upper limit on the sum of the neutrino masses becomes tighter than the ΛCDM limit. Therefore, such models exhibit an even stronger preference for the normal ordering, and their viability could be jeopardized should near-future laboratory experiments determine that the mass ordering is inverted. I then discuss correlations between neutrino and inflationary parameters. I find that our determination of inflationary parameters is relatively stable against reasonable assumptions about the neutrino sector, and thus that neutrino unknowns do not represent an important nuisance for our understanding of inflation and the initial conditions of the Universe. The findings reported in this thesis answer a number of important open questions whose addressing is necessary to ensure a robust detection of neutrino masses (and possibly of the neutrino mass ordering) from future cosmological data, opening the door towards physics beyond the Standard Model.