Leibniz algebras are a non-anticommutative version of Lie algebras. They were introduced by \textit{J.-L. Loday} [Enseign. Math. (2) 39, No. 3--4, 269--293 (1993; Zbl 0806.55009)]. Earlier, they were considered by \textit{A. Blokh} [Sov. Math., Dokl. 6 (1965), 1450--1452 (1966; Zbl 0139.25702); translation from Dokl. Akad. Nauk SSSR 165, 471--473 (1965)] who called them \textit{D-algebras}, for their strict connection with derivations. More precisely, a (left) Leibniz algebra is a vector space \(L\) over a field \(\mathbb{F}\) which satisfies the (left) Leibniz identity, i.e.\ the (left) multiplications are derivations. The aim of this paper is to introduce and study the notions of \textit{prime} and \textit{semi-prime} ideals in the context of Leibniz algebras. The notion of prime ideals plays an important role in the theory of rings and in the theory of associative algebras. Some of the results are still valid in the context of Lie and Leibniz algebras, while some others fail to hold. For instance, a Leibniz algebra \(\mathfrak{g}\) is prime if and only if its \textit{Leibniz kernel} \(\operatorname{Leib}(\mathfrak{g})\) is a prime ideal, while in general the zero ideal of \(\mathfrak{g}\) is not prime. The authors give several characterizations of prime ideals in the context of Leibniz algebras and study relations between \textit{prime}, \textit{semi-prime}, \textit{maximal} and \textit{irreducible} ideals. They prove that, if \(\mathfrak{g}\) satisfies the \textit{maximal condition on ideals}, then any semi-prime ideal of \(\mathfrak{g}\) is the intersection of a finite number of prime ideals. Moreover, the radical of a Leibniz algebra \(\mathfrak{g}\) coincides with the intersection of all semi-prime ideals, which is equal to the intersection of all prime ideals of \(\mathfrak{g}\). Furthermore, they prove that, given any Leibniz algebra \(\mathfrak{g}\) and an ideal \(I \subseteq \mathfrak{g}\), the quotient algebra \(\mathfrak{g}/I\) is prime if and only if \(I\) is a prime ideal of \(\mathfrak{g}\). As a consequence, a Leibniz algebra \(\mathfrak{g}\) is prime if and only if the Lie algebra \(\mathfrak{g}/\operatorname{Leib}(\mathfrak{g})\) is prime.