  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+\epsilon}, 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! ... n_7! \pmod p, where $\max \{n_i |
i=1,... 7\}=O(p^{11/12+\epsilon}), and we find the asymptotic formula for the
number of such representations. Finally, we show that products of 4 factorials
$n_1!n_2!n_3!n_4!, with \max\{n_1, n_2, n_3, n_4\}=O(p^{6/7+\epsilon})$
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+\epsilon})$ are
uniformly distributed modulo p.
