This self-contained introduction to geometric invariant theory links the theory of affine algebraic groups to Mumford's theory. The authors, professors of mathematics at Universidad de la República, Uruguay, exploit the viewpoint of Hopf algebra theory and the theory of comodules to simplify many of the relevant formulas and proofs. Early chapters review prerequisites in commutative algebra, algebraic geometry, and the theory of semisimple Lie algebras. Coverage then progresses from Jordan decomposition through homogeneous spaces and quotients. Chapter exercises, and a glossary, notations, and results are included.