From the reviews: "This book is dealing with Hodge Theory ... which generalizes in a functorial way the variations of MHS. ... The clarity of the presentation and the wealth of information are both remarkable. This book ... is a masterpiece that anyone working in Algebraic Geometry, Singularities or Analytic/Complex Geometry would like to have in his own library." (Alexandru Dimca, Zentralblatt MATH, Vol. 1138 (16), 2008) "The book under review ... focuses mainly on the 'pure' story just summarized, is aimed at graduate students and researchers ... . The book begins with a brief historical survey; each chapter is headed by a good summary of its contents and concluded by historical remarks (with references). ... this work is a thoroughly readable and very up-to-date account of mixed Hodge theory, written by masters of the subject, and will undoubtedly serve as a basic reference for years to come." (Matt Kerr, Mathematical Reviews, Issue 2009 C) "This book has been awaited for many years. ... the book which is now available will certainly rapidly become one of the standard references on the topic. Hodge theory assigns to a complex variety data which come from linear algebra. ... I heartily recommend the book." (Helene Esnault, Jahresbericht der Deutsche Mathematiker Vereinigung, Vol. 112 (1), 2010)

• Basic Hodge Theory.- Compact Kähler Manifolds.- Pure Hodge Structures.- Abstract Aspects of Mixed Hodge Structures.- Mixed Hodge Structures on Cohomology Groups.- Smooth Varieties.- Singular Varieties.- Singular Varieties: Complementary Results.- Applications to Algebraic Cycles and to Singularities.- Mixed Hodge Structures on Homotopy Groups.- Hodge Theory and Iterated Integrals.- Hodge Theory and Minimal Models.- Hodge Structures and Local Systems.- Variations of Hodge Structure.- Degenerations of Hodge Structures.- Applications of Asymptotic Hodge Theory.- Perverse Sheaves and D-Modules.- Mixed Hodge Modules.

From the reviews: "This book is dealing with Hodge Theory ... which generalizes in a functorial way the variations of MHS. ... The clarity of the presentation and the wealth of information are both remarkable. This book ... is a masterpiece that anyone working in Algebraic Geometry, Singularities or Analytic/Complex Geometry would like to have in his own library." (Alexandru Dimca, Zentralblatt MATH, Vol. 1138 (16), 2008) "The book under review ... focuses mainly on the 'pure' story just summarized, is aimed at graduate students and researchers ... . The book begins with a brief historical survey; each chapter is headed by a good summary of its contents and concluded by historical remarks (with references). ... this work is a thoroughly readable and very up-to-date account of mixed Hodge theory, written by masters of the subject, and will undoubtedly serve as a basic reference for years to come." (Matt Kerr, Mathematical Reviews, Issue 2009 C) "This book has been awaited for many years. ... the book which is now available will certainly rapidly become one of the standard references on the topic. Hodge theory assigns to a complex variety data which come from linear algebra. ... I heartily recommend the book." (Helene Esnault, Jahresbericht der Deutsche Mathematiker Vereinigung, Vol. 112 (1), 2010)

• Basic Hodge Theory.- Compact Kähler Manifolds.- Pure Hodge Structures.- Abstract Aspects of Mixed Hodge Structures.- Mixed Hodge Structures on Cohomology Groups.- Smooth Varieties.- Singular Varieties.- Singular Varieties: Complementary Results.- Applications to Algebraic Cycles and to Singularities.- Mixed Hodge Structures on Homotopy Groups.- Hodge Theory and Iterated Integrals.- Hodge Theory and Minimal Models.- Hodge Structures and Local Systems.- Variations of Hodge Structure.- Degenerations of Hodge Structures.- Applications of Asymptotic Hodge Theory.- Perverse Sheaves and D-Modules.- Mixed Hodge Modules.