  These notes are an account of a series of lectures I gave at the LMS-CMI
Research School `Homotopy Theory and Arithmetic Geometry: Motivic and
Diophantine Aspects', in July 2018, at the Imperial College London. The goal of
these notes is to see how motives may be used to enhance cohomological methods,
giving natural ways to prove independence of $\ell$ results for traces and
zeta-functions, and constructions of characteristic classes (as $0$-cycles).
This leads to the Grothendieck-Lefschetz formula, of which we give a new
motivic proof. There are also a few additions to what have been told in the
lectures: a proof of Grothendieck-Verdier duality of \'etale motives on schemes
of finite type over a regular quasi-excellent scheme (which slightly improves
the level of generality in the existing literature); a proof that
$\mathbf{Q}$-linear motivic sheaves are virtually integral; a proof of the
motivic generic base change formula.
