The Erdős-Hajnal conjecture says that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a clique or stable set of size at least $n^c$. In this paper we are concerned with the case when $H$ is a path. The conjecture has been proved for paths with at most five vertices, but not for longer paths. We prove that the conjecture is ``nearly'' true for all paths: for every path $H$, all $H$-free graphs with $n$ vertices have cliques or stable sets of size at least $2^{(\log n)^{1-o(1)}}$.