I. Stability Analysis of Difference Schemes by the Method of Differential Approximation.- 1. Certain Properties of the Theory of Linear Differential Equations and Difference Schemes.- 2. The Concept of the Differential Approximation of a Difference Scheme.- 3. Stability Analysis of Difference Schemes with Constant Coefficients by Means of the Differential Representation.- 4. Connection Between The Stability of Difference Schemes and the Properties of Their First Differential Approximations.- 5. Dissipative Difference Schemes for Hyperbolic Equations.- 6. A Means for the Construction of Difference Schemes with Higher Order of Approximation.- II. Investigation of the Artificial Viscosity of Difference Schemes.- 7. K-property of Difference Schemes.- 8. Investigation of Dissipation and Dispersion of Difference Schemes.- 9. Application of the Method of Differential Approximation to the Investigation of the Effects of Nonlinear Transformations.- 10. Investigation of Monotonicity of Difference Schemes.- 11. Difference Schemes in an Arbitrary Curvilinear Coordinate System.- III. Invariant Difference Schemes.- 12. Some Basic Concepts of the Theory of Group Properties of Differential Equations.- 13. Groups Admitted by the System of the Equations of Gas Dynamics.- 14. A Necessary and Sufficient Condition for Invariance of Difference Schemes on the Basis of the First Differential Approximation.- 15. Conditions for the Invariance of Difference Schemes for the One-dimensional Equations of Gas Dynamics.- 16. Investigation of Properties of the Artificial Viscosity of Invariant Difference Schemes for the One-dimensional Equations of Gas Dynamics.- 17. Conditions for the Invariance of Difference Schemes for the Two-dimensional Equations of Gas Dynamics.- 18. Investigation of Difference Schemes with Time-splitting Using the Theory of Groups.- IV. Appendix.- A.1 Introduction.- A.2 Difference Schemes for the Equation of Propagation.- A.3 Difference Schemes for the Equations of One-dimensional Gas Dynamics.- References.