This two volume work on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Whereas Volume I is more elementary, the present Volume II is more at the research level and somewhat more specialized. Both volumes are also available as hardcover editionas Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete". From the reviews: "The main theme of this ... monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. ... The book is written for mathematicians interested in the modern development of algebraic geometry." (EMS Newsletter, September, 2006) From the reviews: "The main theme of this monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. The book is written for mathematicians interested in the modern development of algebraic geometry." (EMS Newsletter, September, 2006)" From the reviews: "The main theme of this monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. The book is written for mathematicians interested in the modern development of algebraic geometry." (EMS Newsletter, September, 2006)" From the reviews: "The main theme of this monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. The book is written for mathematicians interested in the modern development of algebraic geometry." (EMS Newsletter, September, 2006) From the reviews: "The main theme of this monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. The book is written for mathematicians interested in the modern development of algebraic geometry." (EMS Newsletter, September, 2006) From the reviews: "Positivity has been a major theme in various branches of algebraic geometry a ] . the author succeeds wonderfully in putting together under the same heading most of the areas a ] . the book contains genuinely new results. a ] There are numerous examples a ] . is exceptionally well written. a ] will be of great value to both students and experts in the field. It is also excellent as a guide to further literature. In my opinion, Lazarsfelda (TM)s book will become one of the fundamental references a ] ." (Mihnea Popa, Mathematical Reviews, 2005k) From the reviews: "Positivity has been a major theme in various branches of algebraic geometry ??? . the author succeeds wonderfully in putting together under the same heading most of the areas ??? . the book contains genuinely new results. ??? There are numerous examples ??? . is exceptionally well written. ??? will be of great value to both students and experts in the field. It is also excellent as a guide to further literature. In my opinion, Lazarsfeld??'s book will become one of the fundamental references ??? ." (Mihnea Popa, Mathematical Reviews, 2005k) This two-volume book on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Most of the material in the present Volume II has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. Both volumes are also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete."