  In this paper we study asymptotic properties of families of zeta and
$L$-functions over finite fields. We do it in the context of three main
problems: the basic inequality, the Brauer--Siegel type results and the results
on distribution of zeroes. We generalize to this abstract setting the results
of Tsfasman, Vl\u{a}du\c{t} and Lachaud, who studied similar problems for
curves and (in some cases) for varieties over finite fields. In the classical
case of zeta functions of curves we extend a result of Ihara on the limit
behaviour of the Euler--Kronecker constant. Our results also apply to
$L$-functions of elliptic surfaces over finite fields, where we approach the
Brauer--Siegel type conjectures recently made by Kunyavskii, Tsfasman and
Hindry.
