  We study the Linial--Meshulam model of random two-dimensional simplicial
complexes. One of our main results states that for $p\ll n^{-1}$ a random
2-complex $Y$ collapses simplicially to a graph and, in particular, the
fundamental group $\pi_1(Y)$ is free and $H_2(Y)=0$, a.a.s. We also prove that,
if the probability parameter $p$ satisfies $p\gg n^{-1/2+\epsilon}$, where
$\epsilon>0$, then an arbitrary finite two-dimensional simplicial complex
admits a topological embedding into a random 2-complex, with probability
tending to one as $n\to \infty$. We also establish several related results, for
example we show that for $p<c/n$ with $c<3$ the fundamental group of a random
2-complex contains a nonabelian free subgroup. Our method is based on
exploiting explicit thresholds (established in the paper) for the existence of
simplicial embedding and immersions of 2-complexes into a random 2-complex.
