  In many interesting situations the size of epsilon-nets depends only on
$\epsilon$ together with different complexity measures. The aim of this paper
is to give a systematic treatment of such complexity measures arising in
Discrete and Computational Geometry and Statistical Learning, and to bridge the
gap between the results appearing in these two fields. As a byproduct, we
obtain several new upper bounds on the sizes of epsilon-nets that
generalize/improve the best known general guarantees. In particular, our
results work with regimes when small epsilon-nets of size
$o(\frac{1}{\epsilon})$ exist, which are not usually covered by standard upper
bounds. Inspired by results in Statistical Learning we also give a short proof
of the Haussler's upper bound on packing numbers.
