Neutrinos can play an important role in the evolution of the Universe, modifying some of the cosmological observables. In this contribution we summarize the main aspects of cosmological relic neutrinos and we describe how the precision of present cosmological data can be used to learn about neutrino properties, in particular their mass, providing complementary information to beta decay and neutrinoless double-beta decay experiments. We show how the analysis of current cosmological observations, such as the anisotropies of the cosmic microwave background or the distribution of large-scale structure, provides an upper bound on the sum of neutrino masses of order 1 eV or less, with very good perspectives from future cosmological measurements which are expected to be sensitive to neutrino masses well into the sub-eV range.