Scattering theory investigates the interaction of a propagating wave, or incident wave, with an obstacle, or scatterer. The existence of the scatterer disturbs the incident wave in a way that depends on the physical and geometrical properties of the scatterer. In the forward scattering problem one knows the incident wave, as well as the physical and geometrical characteristics of the scatterer, and seeks the effect that the scatterer has on the propagation of the incident wave. More interesting and much more difficult is the inverse scattering problem, where one again knows the incident wave and has (usually partial) knowledge about the form of disturbance caused by the scatterer, and the goal is to identify as much information as possible about the physics and/or geometry of the scatterer. A large portion of modern science and technology is founded on inverse scattering problems. In this chapter, we state the appropriate boundary value problems for scattering of acoustic, electromagnetic, and elastic waves, then we focus on the special case where the wavelength of the incident wave is much larger than the characteristic dimension of the scatterer, and finally we solve some representative scattering problems when the scatterer is, or can be approximated by, an ellipsoid. The study of small scatterers, as they compare to the wavelength, is known as the theory of low-frequency scattering, and was initiated by Rayleigh [287], formally developed by Stevenson [326], and established into a rigorous mathematical theory by Kleinman [215].