This is the first comprehensive basic monograph on mixed Hodge structures. Starting with a summary of classic Hodge theory from a modern vantage point the book goes on to explain Deligne's mixed Hodge theory. Here proofs are given using cubical schemes rather than simplicial schemes. Next come Hain's and Morgan's results on mixed Hodge structures related to homotopy theory. Steenbrink's approach of the limit mixed Hodge structure is then explained using the language of nearby and vanishing cycle functors bridging the passage to Saito's theory of mixed Hodge modules which is the subject of the last chapter. Since here D-modules are essential, these are briefly introduced in a previous chapter. At various stages applications are given, ranging from the Hodge conjecture to singularities. The book ends with three large appendices, each one in itself a resourceful summary of tools and results not easily found in one place in the existing literature (homological algebra, algebraic and differential topology, stratified spaces and singularities). The book is intended for advanced graduate students, researchers in complex algebraic geometry as well as interested researchers in nearby fields (algebraic geometry, mathematical physics
