The book is based on an annual course of lectures that the authors gave at Moscow State University. A systematic presentation of methods for constructing and studying difference schemes for the numerical solution of one-dimensional transient problems of gas dynamics and magnetic hydrodynamics is given, including the results obtained in recent years. Along with difference schemes, the book pays great attention to the algorithms that implement them. A significant part of the schemes and algorithms described in the book has been tested in practice when solving a large number of different problems. Examples of such calculations are given, illustrating the effectiveness of the selected schemes and algorithms.