Let \(q\) be an odd prime, and let \(\chi\) be a non-principal Dirichlet character modulo \(q\). Define \[ M(\chi)=\max_{1\leq x\leq q}\left|\sum_{n\leq x}\chi(n)\right|, \] and let \(N_{\chi}\) denote a point at which the maximum is attained. This paper proved that the distribution of \(M(\chi)/\sqrt{q}\) converses weakly to a universal distribution \(\Phi\), uniformly throughout most of the possible range, and got estimates for \(\Phi\)’s tail. The paper also showed that for most \(\chi\) with \(M(\chi)\) large, \(N_{\chi}\) is bounded away from \(q/2\), and the value of \(M(\chi)\) is little larger than \(\left(\sqrt{q}/\pi\right)|L(1,\chi)|\).