We consider the direct and inverse scattering problems for the Schrödinger equation of quantum mechanics and for the Helmholtz equation of acoustics and electrodynamics. Moreover, only scattering data without phase information can be directly measured in practice in quantum mechanics and in some other cases. In particular, in quantum mechanics, this restriction is associated with the probabilistic interpretation of the wave function proposed by M. Born in 1926. In this regard, we present results on non-uniqueness, uniqueness, and reconstruction in the inverse scattering problem without phase information. We are motivated by recent and very significant progress in this area.