  This paper is focused on $f$-divergences, consisting of three main
contributions. The first one introduces integral representations of a general
$f$-divergence by means of the relative information spectrum. The second part
provides a new approach for the derivation of $f$-divergence inequalities, and
it exemplifies their utility in the setup of Bayesian binary hypothesis
testing. The last part of this paper further studies the local behavior of
$f$-divergences.
