The analysis of stellar abundances for odd-Z Fe-peak elements requires accurate non-local thermodynamic equilibrium (NLTE) modelling of spectral lines fully taking into account the hyperfine structure (HFS) splitting of lines. Here, we investigate the statistical equilibrium of Co in the atmospheres of cool stars and the influence of NLTE and HFS on the formation of Co lines and abundances. Significant departures from LTE level populations are found for Co i; number densities of excited states in Co ii also differ from LTE at low metallicity. The NLTE level populations are used to determine the abundance of Co in solar photosphere, log ɛ= 4.95 � 0.04 dex, which is in agreement with that in C i meteorites within the combined uncertainties. The spectral lines of Co i were calculated using the results of recent measurements of hyperfine interaction constants by UV Fourier transform spectrometry. For Co ii, the first laboratory measurements of HFS A and B factors were performed. These highly accurate A factor measurements (errors of the order of 3–7 per cent) allow, for the first time, reliable modelling of Co ii lines in the solar and stellar spectra and, thus, a test of the Co i/Co ii ionization equilibrium in stellar atmospheres. A differential abundance analysis of Co is carried out for 18 stars in the metallicity range −3.12 < [Fe/H] < 0. The abundances are derived by the method of spectrum synthesis. At low [Fe/H], NLTE abundance corrections for Co i lines are as large as +0.6, … , + 0.8 dex. Thus, LTE abundances of Co in metal-poor stars are severely underestimated. The stellar NLTE abundances determined from the single UV line of Co ii are lower by ∼0.5–0.6 dex. The discrepancy might be attributed to possible blends that have not been accounted for in the solar Co ii line and its erroneous oscillator strength. The increasing [Co/Fe] trend in metal-poor stars, as calculated from the Co i lines under NLTE, can be explained if Co is overproduced relative to Fe in massive stars. The models of Galactic chemical evolution are wholly inadequate to describe this trend suggesting that the problem is in supernova yields.