Derived categories There have been remarkably few systematic expositions of the theory of derived categories since its inception in the work of Grothendieck and Verdier in the 1960s. This book is the first in depth treatment of this important component of homological algebra. It carefully explains the foundations in detail before moving on to key applications in commutative and noncommutative algebra many otherwise unavailable outside of research articles. These include commutative and noncommutative dualizing complexes perfect DG modules and tilting DG bimodules. Written with graduate students in mind the emphasis here is on explicit constructions (with many examples and exercises) as opposed to axiomatics with the goal of demystifying this difficult subject. Beyond serving as a thorough introduction for students it will serve as an important reference for researchers in algebra geometry and mathematical physics.