This monograph, authored by Vladimir A. Marchenko, focuses on the application of transformation operators in spectral theory and its untraditional applications. The book is divided into four chapters, each addressing different aspects of the Sturm-Liouville equation and operator theory. **Chapter 1** introduces transformation operators and their use in investigating the Sturm-Liouville boundary value problem on a finite interval. It covers the completeness of eigenfunctions and generalized eigenfunctions, asymptotic formulas for solutions and eigenvalues, and the derivation of Gelfand-Levitan trace formulas. **Chapter 2** discusses singular boundary value problems on the half-line, introducing the concept of distribution-valued spectral functions and proving the existence of such functions. It also derives the Gelfand-Levitan integral equation and conditions for a distribution to be a spectral function. The chapter concludes with the asymptotic formula for spectral functions and Levitan's equiconvergence theorem. **Chapter 3** addresses inverse problems in scattering theory and the inverse problem for the Hill equation. It introduces Levin transformation operators and uses them to study solutions to Sturm-Liouville equations with specific constraints. The chapter derives Marchenko's integral equation, establishes properties of scattering data, and discusses results on stability zones and the inverse problem for the full real line. **Chapter 4** explores the use of spectral theory in integrating certain nonlinear partial differential equations, focusing on the Korteweg-de Vries equation. It presents a new integration method, solves the Cauchy problem for the Korteweg-de Vries equation, and discusses the method developed by V. E. Zakharov and A. B. Shabat. The chapter also includes exercises that guide readers towards generalizations and further research. The author emphasizes that the book does not cover all aspects of spectral theory but aims to provide a comprehensive discussion of transformation operators and their applications.This monograph, authored by Vladimir A. Marchenko, focuses on the application of transformation operators in spectral theory and its untraditional applications. The book is divided into four chapters, each addressing different aspects of the Sturm-Liouville equation and operator theory. **Chapter 1** introduces transformation operators and their use in investigating the Sturm-Liouville boundary value problem on a finite interval. It covers the completeness of eigenfunctions and generalized eigenfunctions, asymptotic formulas for solutions and eigenvalues, and the derivation of Gelfand-Levitan trace formulas. **Chapter 2** discusses singular boundary value problems on the half-line, introducing the concept of distribution-valued spectral functions and proving the existence of such functions. It also derives the Gelfand-Levitan integral equation and conditions for a distribution to be a spectral function. The chapter concludes with the asymptotic formula for spectral functions and Levitan's equiconvergence theorem. **Chapter 3** addresses inverse problems in scattering theory and the inverse problem for the Hill equation. It introduces Levin transformation operators and uses them to study solutions to Sturm-Liouville equations with specific constraints. The chapter derives Marchenko's integral equation, establishes properties of scattering data, and discusses results on stability zones and the inverse problem for the full real line. **Chapter 4** explores the use of spectral theory in integrating certain nonlinear partial differential equations, focusing on the Korteweg-de Vries equation. It presents a new integration method, solves the Cauchy problem for the Korteweg-de Vries equation, and discusses the method developed by V. E. Zakharov and A. B. Shabat. The chapter also includes exercises that guide readers towards generalizations and further research. The author emphasizes that the book does not cover all aspects of spectral theory but aims to provide a comprehensive discussion of transformation operators and their applications.