The polarization and strain response of ferroelectric materials at electric fields below the macroscopic coercive field is of paramount importance for the operation of many electronic devices. The response of real ferroelectric and related materials is, in general, complex and difficult to interpret. The reason for this is that many processes in a ferroelectric material contribute to its properties, often concurrently. Examples include the motion of ferroelectric and ferroelastic domains, the presence of domains within domains, the dynamics of different types of polar nano-entities, the interaction of polar nano-entities (e.g., polar nanoregions in relaxors) with the strain and polarization within domains, motion of defects, and rearrangement of defect clusters and their interaction with polarization and strain. One signature of these processes is nonlinearity of the strain and polarization. Most ferroelectrics exhibit nonlinear response at all practical field levels, meaning that the apparent material coefficients depend on the amplitude of the driving excitation. In this paper, we show that an investigation of nonlinear behavior is a sensitive way to study various mechanisms operating in dielectric and piezoelectric materials. We review the basic formalism of the nonlinear description of polarization and strain, give a physical interpretation of different terms, and illustrate this approach on numerous examples of relaxors, relaxor ferroelectrics, hard and soft ferroelectrics, and morphotropic phase boundary compositions. An experimental approach based on a lock-in technique that is well suited for such studies is also discussed.