  Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the
mixing time is upper bounded by the maximum hitting time of the graph. This
fails for directed chains, as the biased random walk on the cycle
$\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the
mixing time is $O(mn)$, where $m$ is the number of edges and $n$ is the number
of vertices. In the reversible case, the mixing time is robust to the change of
the laziness parameter. Surprisingly, in the directed setting the mixing time
can be sensitive to such changes. We also study exploration and cover times for
random walks on Eulerian digraphs and prove universal upper bounds in analogy
to the undirected case.
