Complex algebraic threefolds The first book on the explicit birational geometry of complex algebraic threefolds this detailed text covers all the knowledge of threefolds needed to enter the field of higher dimensional birational geometry. Containing over 100 examples and many recent results it is suitable for advanced graduate students as well as researchers The first book on the explicit birational geometry of complex algebraic threefolds arising from the minimal model program this text is sure to become an essential reference in the field of birational geometry. Threefolds remain the interface between low and high dimensional settings and a good understanding of them is necessary in this actively evolving area. Intended for advanced graduate students as well as researchers working in birational geometry the book is as self contained as possible. Detailed proofs are given throughout and more than 100 examples help to deepen understanding of birational geometry. The first part of the book deals with threefold singularities divisorial contractions and flips. After a thorough explanation of the Sarkisov program the second part is devoted to the analysis of outputs specifically minimal models and Mori fibre spaces. The latter are divided into conical fibrations del Pezzo fibrations and Fano threefolds according to the relative dimension.