Homogeneous spaces of algebraic groups naturally arise in various problems of geometry and representation theory. The same reasons that motivate considering projective spaces instead of affine spaces (e.g. solutions “at infinity” of systems of algebraic equations) stimulate the study of compactifications or, more generally, equivariant embeddings of homogeneous spaces. The embedding theory of a homogeneous space is governed by a certain numerical invariant called complexity. We discuss the geometric and representation-theoretic meaning and methods to compute this invariant. Homogeneous spaces of complexity zero are called spherical. They can be characterized by a number of remarkable equivalent conditions and have an elegant and well controlled theory of equivariant embeddings. We consider various applications of this theory. As a particular case, we study equivariant embeddings of reductive groups. The embedding theory of spherical spaces is deduced from general results of Luna and Vust on embeddings of arbitrary homogeneous spaces and can be generalized to homogeneous spaces of the “next level of complexity”—complexity one.