  In this paper we consider the problem of embedding almost-spanning, bounded
degree graphs in a random graph. In particular, let $\Delta\geq 5$,
$\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and
with maximum degree $\Delta$. We show that a random graph $G_{n,p}$ with high
probability contains a copy of $H$, provided that $p\gg
(n^{-1}\log^{1/\Delta}n)^{2/(\Delta+1)}$. Our assumption on $p$ is optimal up
to the $polylog$ factor. We note that this $polylog$ term matches the
conjectured threshold for the spanning case.
