  Inverse scattering and spectral one-dimensional problems are discussed
systematically in a self-contained way. Many novel results, due to the author
are presented. The classical results are often presented in a new way. Several
highlights of the new results include:
  Analysis of the invertibility of the steps in the Gel'fand-Levitan and
Marchenko inversion procedures, Theory of the inverse problem with I-function
as the data and its applications; Proof of the property C for ordinary
differential operators, numerous applications of property C; Inverse problems
with "incomplete" data; Spherically symmetric inverse scattering problem with
fixed-energy data: analysis of the Newton-Sabatier (NS) scheme for inversion of
fixed-energy phase shifts is given. This analysis shows that the NS scheme is
fundamentally wrong, and is not a valid inversion method.
  Complete presentation of the Krein inverse scattering theory is given.
Consistency of this theory is proved. Quarkonium systems; Some new inverse
problems for the heat and wave equations. A study of inverse scattering problem
for an inhomogeneous Schr\"odinger equation;
