  Property C stands for completeness of the set of products of solutions to
homogeneous linear differential equations. property C is proved in various
formulations for Schr\"odinger operators. Many applications of this property to
inverse problems and inverse scattering problems are given. It is shown what
part of the fixed-energy phase shifts determines a compactly supported
potential uniquely. It is shown that the Newton-Sabatier method is not really
an inversion method but a parameter-fitting procedure and it is proved that
this procedure cannot recover a generic potential, in particular, any potential
which is not analytic in a neighborhood of the positive real axis.
