  We show that the largest disc covered by a simple random walk (SRW) on
$\mathbb{Z}^2$ after n steps has radius n^{1/4+o(1)}, thus resolving an open
problem of R\'{e}v\'{e}sz [Random Walk in Random and Non-Random Environments
(1990) World Scientific, Teaneck, NJ]. For any fixed $\ell$, the largest disc
completely covered at least $\ell$ times by the SRW also has radius
n^{1/4+o(1)}. However, the largest disc completely covered by each of $\ell$
independent simple random walks on $\mathbb{Z}^2$ after $n$ steps is only of
radius $n^{1/(2+2\sqrt{\ell})+o(1)}$. We complement this by showing that the
radius of the largest disc completely covered at least a fixed fraction
$\alpha$ of the maximum number of visits to any site during the first $n$ steps
of the SRW on $\mathbb{Z}^2$, is $n^{(1-\sqrt{\alpha})/4+o(1)}$. We also show
that almost surely, for infinitely many values of $n$ it takes about
$n^{1/2+o(1)}$ steps after step n for the SRW to reach the first previously
unvisited site (and the exponent 1/2 is sharp). This resolves a problem raised
by R\'{e}v\'{e}sz [Ann. Probab. 21 (1993) 318--328].
