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The AMS now makes available this succinct and quite elegant research monograph written by Fields Medalist and eminent researcher, Laurent Lafforgue. The material is an outgrowth of Lafforgue's lectures and seminar at the Centre de Recherches Mathematiques (University of Montreal, QC, Canada), where he held the 2001-2002 Aisenstadt Chair. In the book, he addresses an important recurrent theme of modern mathematics: the various compactifications of moduli spaces, which have a large number of applications.This book treats the case of thin Schubert varieties, which are natural subvarieties of Grassmannians. He was led to these questions by a particular case linked to his work on the Langlands program. In this monograph, he develops the theory in a more systematic way, which exhibits strong similarities with the case of moduli of stable curves. Prerequisites are minimal and include basic algebraic geometry, and standard facts about Grassmann varieties, their Plucker embeddings, and toric varieties. The book is suitable for advanced graduate students and research mathematicians interested in the classification of moduli spaces.