The book provides a comprehensive overview on the theory on analysis of singularities for partial differential equations (PDEs). It contains a summarization of the formation, development and main results on this topic. Some of the author's discoveries and original contributions are also included, such as the propagation of singularities of solutions to nonlinear equations, singularity index and formation of shocks.Contents:Introduction to Problems on Singularity AnalysisSingularity Analysis for Linear EquationsSingularity Analysis for Semilinear EquationsPropagation of Singularities for Fully Nonlinear EquationsPropagation of Strong Singularities for Nonlinear EquationsFormation of Shocks for Quasilinear Hyperbolic EquationsReadership: Graduate students and researchers interested in analysis and differential equations.