  We determine the permutation groups
$P_{\mathrm{comp}}(\mathbb{F}_q),P_{\mathrm{orth}}(\mathbb{F}_q)\leq\operatorname{Sym}(\mathbb{F}_q)$
generated by the complete mappings, respectively the orthomorphisms, of the
finite field $\mathbb{F}_q$ -- both are equal to
$\operatorname{Sym}(\mathbb{F}_q)$ unless $q\in\{2,3,4,5,8\}$. More generally,
denote by $P_{\mathrm{comp}}(G)$, respectively $P_{\mathrm{orth}}(G)$, the
subgroup of $\operatorname{Sym}(G)$ generated by the complete mappings,
respectively the orthomorphisms, of the group $G$. Using recent results of
Eberhard-Manners-Mrazovi\'c and M\"uyesser-Pokrovskiy, we show that for each
large enough finite group $G$ that has a complete mapping (i.e., whose Sylow
$2$-subgroups are trivial or noncyclic),
$P_{\mathrm{comp}}(G)=\operatorname{Sym}(G)$ and
$P_{\mathrm{orth}}(G)\geq\operatorname{Alt}(G)$. We also prove that
$P_{\mathrm{orth}}(G)=\operatorname{Sym}(G)$ for every large enough finite
solvable group $G$ that has a complete mapping. Proving these results requires
us to study the parities of complete mappings and of orthomorphisms. Some
connections with known results in cryptography and with parity types of Latin
squares are also discussed.
