1 Introduction. Foundations of the Science of Combustion: Basic Physical Concepts.- 1. Some Results from Chemical Kinetics and Thermochemistry.- 2. Self-Accelerating Chemical Reactions and Critical Phenomena - Explosions and Self-Ignition. Semenov's Theory.- 3. The Homogeneous Ideally-Mixing Chemical Reactor.- 4. Laminar Flames.- 2 The Time-Independent Theory of Thermal Explosions.- 1. Basic Assumptions of the Theory.- 2. The Conditions for a Thermal Explosion.- 3. The Stability of the Solutions in the Time-Independent Theory of Thermal Explosions.- 4. Some General Properties of Solutions from the Time-Independent Theory of Thermal Explosions for Vessels of Arbitrary Shape.- 5. Numerical Studies of Thermal Explosions. Experiments.- 6. Application of the Idea of Thermal Explosions to the Mechanics of Polymers.- 3 The Initiation of Chemical Reaction Waves in Fuel Mixtures: Time-Dependent Statements of the Problem.- 1. Ignition of a Fuel Mixture by Heated Surfaces.- 2. The Time-Independent Theory of Ignition by Hot Surfaces as an Intermediate Asymptote of a Time-Dependent Process.- 3. The Initiation of Chemical Reactions by Active Centers.- 4 Laminar Flames.- 1. The Equations of the Theory of Flame Propagation.- 2. Existence and Uniqueness of the Steady-State Solution of the System of Flame Propagation Equations.- 3. The velocity and Structure of a Flame Front for Reactions with Large Activation Energies.- 4. The Spectrum of Flame Propagation Velocities for a Nonzero Reaction Rate near and in the Initial State, The Propagation of Chain Isothermal Flames.- 5. Formation of a Stationary Laminar Flame Front.- 6. The Diffusional-Thermal Stability of Laminar Flames.- 7. Flames in Flows with a Velocity Gradient. Flame "Stretching".- 8. Self-Igniting Mixtures.- 5 Complexand Chain Reactions in Flames.- 1. The Theory of Flames with Multi-step Reactions: Purpose and Methods.- 2. Flames with Unbranched Chain Reactions.- 3. Flames with Branching Chain Reactions.- 4. Simple Bulk-Schemes for Multi-step Chemical Reactions in Flames.- 5. Cool (Isothermal) Flames.- 6 The Gas Dynamics of Combustion.- 1. Motion of Flames in Gas Flows. The Condition for Stationary Combustion.- 2. Combustion in Closed Vessels. The Mach Effect.- 3. Hydrodynamic Instability of Flames.- 4. Flame Acceleration and Detonation in Tubes.- 7 Diffusional Combustion of Gases.- 1. The General Properties of Diffusion Flames. The Combustion Surface.- 2. The Burke-Schumann Problem.- 3. The Limit of Diffusional Combustion in Unmixed Gases.- 4. Diffusion Flames in Opposed Jets of Oxidant and Fuel.