  This note is about the spectral properties of transfer operators associated
to smooth hyperbolic dynamics. In the first two sections (written in 2006), we
state our new results relating such spectra with dynamical determinants, first
announced at the conference ``Traces in Geometry, Number Theory and Quantum
Fields" at the Max Planck Institute, Bonn, October 2005. In the last two
sections, we give a reader-friendly presentation of some key ideas in our work
in the simplest possible settings, including a new proof of a result of Ruelle
on expanding endomorphisms. (These last two sections are a revised version of
the lecture notes given during the workshop ``Resonances and Periodic Orbits:
Spectrum and Zeta functions in Quantum and Classical Chaos" at Institut Henri
Poincar\'e, Paris, July 2005.)
  (Revised version, submitted for publication)
