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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 openproblem 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 disccompletely covered at least $\ell$ times by the SRW also has radiusn^{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 ofradius $n^{1/(2+2\sqrt{\ell})+o(1)}$. We complement this by showing that theradius of the largest disc completely covered at least a fixed fraction$\alpha$ of the maximum number of visits to any site during the first $n$ stepsof the SRW on $\mathbb{Z}^2$, is $n^{(1-\sqrt{\alpha})/4+o(1)}$. We also showthat 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 previouslyunvisited site (and the exponent 1/2 is sharp). This resolves a problem raisedby R\'{e}v\'{e}sz [Ann. Probab. 21 (1993) 318--328].

How large a disc is covered by a random walk in n steps?