Abstract The linear response theory is a powerful tool to study properties of quantum many-particle systems. It uses a one-body perturbation field (as a pump) and measures a change in another (or the same) one-body observable (as a probe). In the context of time-dependent density functional theory (TDDFT), the primary target of the linear response is the density response. The finite amplitude method (FAM) significantly reduces both computational cost and coding tasks, and succeeds in facilitating the linear response calculations in a variety of fields in nuclear physics. We start with a review of the linear response theory and present the essence of the FAM theory and various applications, especially in studies of nuclear collective dynamics.