Let \(\mathcal{A}\) be a Leibniz algebra, \(\operatorname{Der}(\mathcal{A})\) its derivation algebra, and \(\mathrm{Leib}(\mathcal{A})=\langle [x,x]|\ x\in \mathcal{A}\rangle\) the Leibniz kernel of \(\mathcal{A}\). Clearly \(\mathrm{Leib}(\mathcal{A})\) is an abelian ideal of \(\mathcal{A}\) which is also the smallest ideal such that the quotient \(\mathcal{A}/\mathrm{Leib}(\mathcal{A})\) is a Lie algebra. An ideal \(B\) of \(\mathcal{A}\) is said to be characteristic if \(\delta(B)\subseteq B\) for all \(\delta\in Der(\mathcal{A})\). Also, \(\mathcal{A}\) is called complete if the center \(Z(\mathcal{A}/\mathrm{Leib}(\mathcal{A}))=0\) and all derivations of \(\mathcal{A}\) are inner. Clearly if \(\mathcal{A}\) is a Lie algebra, then this definition gives the notion of complete Lie algebra, since in this case \(\mathrm{Leib}(\mathcal{A})=0\). Recall that a subgroup of a group \(G\) is characteristic if it is mapped to itself by every automorphism of \(G\). Also, \(G\) is said to be complete if \(Z(G)=1\) and all automorphisms of \(G\) are inner. For instance, all the symmetric groups \(S_n\) (except when \(n=2,6\)) are complete. In this paper, the authors discuss some results on the structure of characteristic ideals of Leibniz algebras. Also as in the case of Lie algebras, it is shown that (non-zero) nilpotent Leibniz algebras are not complete, but semisimple Leibniz algebras over a field of characteristic zero are complete (\(\mathcal{A}\) is semisimple if \(\mathrm{rad}(\mathcal{A})=\mathrm{Leib}(\mathcal{A})\) where \(\mathrm{rad}(\mathcal{A})\) is the maximal solvable ideal of \(\mathcal{A}\)). Furthermore, it is defined the holomorph of a Leibniz algebra and given necessary and sufficient conditions for its completeness. Note that in [\textit{J. Q. Adashev} et al., Linear Multilinear Algebra 69, 1500--1520 (2021; Zbl 1475.17003)], a complete Leibniz algebra is defined to be same as in the case of Lie algebra with the inner derivation referring to right multiplication. As a difference, by this definition a semisimple Leibniz algebra over a field of characteristic zero is not necessarily complete, but will be complete by the definition of completeness in the paper under review.