The basic experimental data and the theory for autowave processes in active kinetic systems are reviewed. Each volume element in such a system is in a state far from thermodynamic equilibrium, and the different volume elements are coupled by transport processes. Some examples of these systems are certain chemical and biological objects in which various types of waves and stable structures can be produced. Mathematically, autowave processes are described by quasilinear and nonlinear parabolic equations. These autowave processes are quite different from processes which occur in conservative systems, e.g., solitons. A classification of autowave processes is offered, and the experimental data are summarized. In accordance with this classification, the review itself is organized in sections on the physics of the basic models for autowave systems in a one-dimensional space and qualitative methods for studying them. The basic cases are wave propagation, autonomous wave sources, spontaneous oscillations and quasistochastic waves which are synchronized over the entire space, and the formation of dissipative structures. At present, the primary fields of application of the theory of autowave processes are neural conductivity, combustion, self-organization in living systems, etc. The necessary conditions for these autowave situations are listed.