  Methods to determine abundances of the elements in the atmospheres of F, G,
and K stars with precisions of 0.01-0.03 dex are reviewed. Such high-precision
stellar abundances have led to the discovery of unexpected phenomena and
relations with important bearings on the astrophysics of galaxies, stars, and
planets, such as existence of discrete stellar populations in the Galactic disk
and halo, differences in the relation between abundances and elemental
condensation temperature for the Sun and solar twins, differences in chemical
composition between binary star components and between members of star
clusters, and tight relations between some abundance ratios and age for
solar-like stars. We conclude that if high-precision abundances can be achieved
in studies of more distant stars and stars on the giant and supergiant
branches, many more interesting future applications, of great relevance to
stellar and galaxy evolution, will be possible. Hence, in planning abundance
surveys, it is important to carefully balance the need for large samples of
stars against the spectral resolution and signal-to-noise ratio needed to
obtain high-precision abundances. Furthermore, it is an advantage to work
differentially on stars with similar atmospheric parameters, because then a
simple 1D LTE analysis of stellar spectra may be sufficient. However, when
determining high-precision absolute abundances or differential abundances
between stars having more widely different parameters, e.g. metal-poor stars
compared to the Sun or giants to dwarfs, then 3D non-LTE effects must be taken
into account.
