  This is a draft of a monograph to appear in the Springer series
"Encyclopaedia of Mathematical Sciences", subseries "Invariant Theory and
Algebraic Transformation Groups". The subject is homogeneous spaces of
algebraic groups and their equivariant embeddings. The style of exposition is
intermediate between survey and detailed monograph: some results are supplied
with detailed proofs, while the other are cited without proofs but with
references to the original papers.
  The content is briefly as follows. Starting with basic properties of
algebraic homogeneous spaces and related objects, such as induced
representations, we focus the attention on homogeneous spaces of reductive
groups and introduce two important invariants, called complexity and rank. For
the embedding theory it is important that homogeneous spaces of small
complexity admit a transparent combinatorial description of their equivariant
embeddings. We consider the Luna-Vust theory of equivariant embeddings, paying
special attention to the case of complexity not greater than one. A special
chapter is devoted to spherical varieties (= embeddings of homogeneous spaces
of complexity zero), due to their particular importance and ubiquity. A
relation between equivariant embedding theory and equivariant symplectic
geometry is also discussed. The book contains several classification results
(homogeneous spaces of small complexity, etc).
  The text presented here is not in a final form, and the author will be very
grateful to any interested reader for his comments and/or remarks, which may be
sent to the author by email.
