Atomic nuclei are complex systems of protons and neutrons that strongly interact with each other via an attractive and short-range force, leading to a pattern of dominantly monopole and quadrupole correlations between like particles (i.e., proton-proton and neutron-neutron correlations) in low-lying states of atomic nuclei. Among many nucleon pairs, very few nucleon pairs such as proton and neutron pairs with spin zero, two, and occasionally isoscalar proton-neutron pairs with spin aligned, play a dominant role in low-energy nuclear structure. Therefore the nucleon-pair approximation provides us with an efficient truncation scheme of the full shell model configurations which are otherwise too large to handle for medium and heavy nuclei. Furthemore, the nucleon-pair approximation leads to simple pictures in physics, as the dimension of nucleon-pair subspace is small. In this talk I would like to give a brief review of its history, formulation, validity, applications, as well as its link to previous approaches. Numerical calculations of low-lying states for realistic atomic nuclei are demonstrated with examples. Applications of pair approximations to other problems are also discussed.