The book is about reaction-diffusion equations in unbounded domains with a special emphasis on traveling waves and their generalizations as well as on different notions of propagation. It includes a general presentation of all the classical results in this area. Even for some well known results, in some cases, original proofs are included which are simpler and more elegant than the known ones. The book gives a fairly comprehensive and coherent account of the recent developments and current research in this active area. It also contains some of the basic results about elliptic and parabolic partial differential equations and a chapter on the different versions of the maximum principles. Thus, it also serves as an introduction to these topics. Each chapter is made as much autonomous as possible. Each one has a specific introduction as well as brief mentions of extensions or of related subjects. Some outstanding open problems are mentioned along the way. Each introduction states the goals of the chapter, some of its main results, the framework and indicates how the chapter is organized. The book is addressed to researchers and graduate students in mathematics, in particular in analysis, partial differential equations and applied mathematics. It will be of interest as well to researchers and graduate students concerned by mathematical modeling in physics and in biology. It is planed to be a reference book of lasting value with all the important results on a topic which is commonly used in these fields.