Homogeneous Spaces and Equivariant Embeddings Homogeneous spaces of linear algebraic groups lie at the crossroads of algebraic geometry theory of algebraic groups classical projective and enumerative geometry harmonic analysis and representation theory. By standard reasons of algebraic geometry in order to solve various problems on a homogeneous space it is natural and helpful to compactify it while keeping track of the group action i.e. to consider equivariant completions or more generally open embeddings of a given homogeneous space. Such equivariant embeddings are the subject of this book. We focus on the classification of equivariant embeddings in terms of certain data of combinatorial nature (the Luna Vust theory) and description of various geometric and representation theoretic properties of these varieties based on these data. The class of spherical varieties intensively studied during the last three decades is of special interest in the scope of this book. Spherical varieties include many classical examples such as Grassmannians flag varieties and varieties of quadrics as well as well known toric varieties. We have attempted to cover most of the important issues including the recent substantial progress obtained in and around the theory of spherical varieties. Introduction. 1 Algebraic Homogeneous Spaces 2 Complexity and Rank 3 General Theory of Embeddings 4 Invariant Valuations 5 Spherical Varieties Appendices Bibliography Indices.