The paper presents some new results on the sequence \(n!\pmod p\). Let \({\mathcal A}_p=\{n! \pmod p, n\in [0,p-1]\}\). The authors show that \(\#{\mathcal A}_p\ge ({\sqrt{2}}+o(1)){\sqrt{p}}\) as \(p\to\infty\). This improves the result \(\#{\mathcal A}_p\ge ({\sqrt{41/24}}+o(1)){\sqrt{p}}\) for \(p\to\infty\) of V. C. García [Bol. Soc. Mat. Mex., III. Ser. 13, No. 1, 1–6 (2007; Zbl 1195.11010)]. The authors also study a short interval version of the problem. Let \({\mathcal A}_N=\{n!: L\le n\le L+N\}\). They show that if \(N\gg p^{7/8}\log p\), then \(\#{\mathcal A}_N\ge (1+o(1)){\sqrt{p}}\) as \(p\to\infty\). Finally, they revisit a result from M. Z. Garaev et al. [Trans. Am. Math. Soc. 356, No. 12, 5089–5102 (2004; Zbl 1060.11046)] where it was shown that for all integers \(a\) there is a solution to the congruence \(a\equiv n_1! n_2!\cdots n_7!\pmod p\) with \(n_0:=\max\{n_i: 1\le i\le 7\}=O(p^{11/12+\varepsilon})\) as \(p\to\infty\) and improve the exponent \(11/12\) to \(6/7\).