  We determine an asymptotic formula for the number of labelled 2-connected
(simple) graphs on $n$ vertices and $m$ edges, provided that $m-n\to\infty$ and
$m=O(n\log n)$ as $n\to\infty$. This is the entire range of $m$ not covered by
previous results. The proof involves determining properties of the core and
kernel of random graphs with minimum degree at least 2. The case of
2-edge-connectedness is treated similarly. We also obtain formulae for the
number of 2-connected graphs with given degree sequence for most (`typical')
sequences. Our main result solves a problem of Wright from 1983.
