for random walk on a graph, the cover time $t$ is the number of steps required to visit every vertex at least once. we prove the following large deviation bound for the cover time: for every $a>0$ and $d>0$ there is a constant $c>0$ such that for all $n$ and graphs on $n$ vertices of maximum degree $d$, we have $p(t < an) < \exp(-cn)$. \\ \noindent joint work with itai benjamini and ori gurel-gurevich.