Summary: A forest \({\mathcal F}(n,M)\) chosen uniformly from the family of all labelled unrooted forests with \(n\) vertices and \(M\) edges is studied. We show that, like the Erdős-Rényi random graph \(G(n,M)\), the random forest exhibits three modes of asymptotic behaviour: subcritical, nearcritical and supercritical, with the phase transition at the point \(M= n/2\). For each of the phases, we determine the limit distribution of the size of the \(k\)th largest component of \({\mathcal F}(n,M)\). The similarity to the random graph is far from being complete. For instance, in the supercritical phase, the giant tree in \({\mathcal F}(n,M)\) grows roughly two times slower than the largest component of \(G(n,M)\) and the second largest tree in \({\mathcal F}(n,M)\) is of the order \(n^{2/3}\) for every \(M= n/2+ s\), provided that \(s^ 3 n^{-2}\to \infty\) and \(s= o(n)\), while its counterpart in \(G(n,M)\) is of the order \(n^ 2 s^{-2}\log(s^ 3 n^{-2})\ll n^{2/3}\).