There is a renewed interest in constraining the sum of the masses of the three neutrino flavours by using cosmological measurements. Solar, atmospheric, and reactor neutrino experiments have confirmed neutrino oscillations, implying that neutrinos have non-zero mass, but without pinning down their absolute masses. While it is established that the effect of light neutrinos on the evolution of cosmic structure is small, the upper limits derived from large-scale structure could help significantly to constrain the absolute scale of the neutrino masses. It is also important to know the sum of neutrino masses as it is degenerate with the values of other cosmological parameters, e.g. the amplitude of fluctuations and the primordial spectral index. A summary of cosmological neutrino mass limits is given. Current results from cosmology set an upper limit on the sum of the neutrino masses of ~1 eV, somewhat depending on the data sets used in the analyses and assumed priors on cosmological parameters. It is important to emphasize that the total neutrino mass (`hot dark matter) is derived assuming that the other components in the universe are baryons, cold dark matter and dark energy. We assess the impact of neutrino masses on the matter power spectrum, the cosmic microwave background, peculiar velocities and gravitational lensing. We also discuss future methods to improve the mass upper limits by an order of magnitude.
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