We have performed a thorough analysis of the constraints which can be put on neutrino parameters from cosmological observations, most notably those from the WMAP satellite and the 2dF galaxy survey. For these data we find an upper limit on the sum of active neutrino mass eigenstates of ∑m_ν ≤ 1.0 eV (95% conf.), but this limit is dependent on priors. We find that the WMAP and 2dF data alone cannot rule out the evidence from neutrinoless double beta decay reported by the Heidelberg–Moscow experiment. In terms of the relativistic energy density in neutrinos or other weakly interacting species we find, in units of the equivalent number of neutrino species, N_ν, that N_ν  =  4.0^+3.0_−2.1 (95% conf.). When BBN constraints are added, the bound on N_ν is 2.6^+0.4_−0.3 (95% conf.), suggesting that N_ν could possibly be lower than the standard model value of 3. This can, for instance, be the case in models with very low reheating temperature and incomplete neutrino thermalization. Conversely, if N_ν is fixed to 3 then the data from WMAP and 2dFGRS predict that 0.2458 ≤ Y_P ≤ 0.2471 (95% conf.), which is significantly higher than the observationally measured value. The limit on relativistic energy density changes when a small ν_e chemical potential is present during BBN. In this case the upper bound on N_ν from WMAP, 2dFGRS and BBN is N_ν ≤ 6.5. Finally, we find that a non-zero ∑m_ν can be compensated by an increase in N_ν. One result of this is that the LSND result is not yet ruled out by cosmological observations.