We determine the permutation groups Pcomp(Fq),Porth(Fq)≤Sym(Fq) generated by the complete mappings, respectively the orthomorphisms, of the finite field Fq – both are equal to Sym(Fq) unless q∈{2,3,4,5,8}. More generally, denote by Pcomp(G), respectively Porth(G), the subgroup of Sym(G) generated by the complete mappings, respectively the orthomorphisms, of the group G. Using recent results of Eberhard-Manners-Mrazović and Müyesser-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), Pcomp(G)=Sym(G) and Porth(G)≥Alt(G). We also prove that Porth(G)=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.