  This book discusses the construction of triangulated categories of mixed
motives over a noetherian scheme of finite dimension, extending Voevodsky's
definition of motives over a field. In particular, it is shown that motives
with rational coefficients satisfy the formalism of the six operations of
Grothendieck. This is achieved by studying descent properties of motives, as
well as by comparing different presentations of these categories, following and
extending insights and constructions of Deligne, Beilinson, Bloch, Thomason,
Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r,
and others. In particular, the relation of motives with $K$-theory is addressed
in full, and we prove the absolute purity theorem with rational coefficients,
using Quillen's localization theorem in algebraic $K$-theory together with a
variation on the Grothendieck-Riemann-Roch theorem. Using resolution of
singularities via alterations of de Jong-Gabber, this leads to a version of
Grothendieck-Verdier duality for constructible motivic sheaves with rational
coefficients over rather general base schemes. We also study versions with
integral coefficients, constructed via sheaves with transfers, for which we
obtain partial results. Finally, we associate to any mixed Weil cohomology a
system of categories of coefficients and well behaved realization functors,
establishing a correspondence between mixed Weil cohomologies and suitable
systems of coefficients. The results of this book have already served as ground
reference in many subsequent works on motivic sheaves and their realizations,
and pointers to the most recent developments of the theory are given in the
introduction.
