  We present a unified general method for the asymptotic study of graphs from
the so-called "subcritical"$ $ graph classes, which include the classes of
cacti graphs, outerplanar graphs, and series-parallel graphs. This general
method works both in the labelled and unlabelled framework. The main results
concern the asymptotic enumeration and the limit laws of properties of random
graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp.
$g_n$) of labelled (resp. unlabelled) graphs on $n$ vertices from a subcritical
graph class ${G}=\cup_n {G_n}$ satisfies asymptotically the universal behaviour
$$ g_n = c n^{-5/2} \gamma^n (1+o(1)) $$ for computable constants $c,\gamma$,
e.g. $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that
the number of vertices of degree $k$ ($k$ fixed) in a graph chosen uniformly at
random from $G_n$, converges (after rescaling) to a normal law as $n\to\infty$.
