This book combines foundational constructions in the theory of motives and results relating motivic cohomology to more explicit constructions. Prerequisite for understanding the work is a basic background in algebraic geometry. The author constructs and describes a triangulated category of mixed motives over an arbitrary base scheme. Most of the classical constructions of cohomology are described in the motivic setting, including Chern classes from higher $K$-theory, push-forward for proper maps, Riemann-Roch, duality, as well as an associated motivic homology, Borel-Moore homology and cohomology with compact supports.
Motives: Introduction: Part I The motivic category Motivic cohomology and higher Chow groups K-theory and motives Homology, cohomology and duality Realization of the motivic category Motivic constructions and comparisons Equidimensional cycles K-theory Categorical algebra: Introduction: Part II Symmetric monoidal structures DG categories and triangulated categories Simplicial and cosimplicial constructions Canonical models for cohomology Bibliography Subject index Index of notation.