Character sums and congruences with $n!$

We estimate character sums with $n!$, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime $p$ and obtain new information about the spacings between quadratic nonresidues modulo $p$. In particular, we show that there exists a positive integer $n\ll p^{1/2+\varepsilon}$ such that $n!$ is a primitive root modulo $p$. We also show that every nonzero congruence class $a \not \equiv 0 \pmod p$can be represented as a product of 7 factorials, $a \equiv n_1! \ldots n_7! \pmod p$, where $\max \{n_i \vert i=1,\ldots, 7\}=O(p^{11/12+\varepsilon})$, and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials $n_1!n_2!n_3!n_4!,$ with $\max\{n_1, n_2, n_3, n_4\}=O(p^{6/7+\varepsilon})$ represent ``almost all'' residue classes modulo p, and that products of 3 factorials $n_1!n_2!n_3!$ with $\max\{n_1, n_2, n_3\}=O(p^{5/6+\varepsilon})$ are uniformly distributed modulo $p$.