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Abstract. We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, a priori, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass M of the field and the cosmological constant Λ > 0 : it appears a critical value, Λ/12, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When M 2 ≥ Λ/12 there exists a unique unitary dynamics. In opposite, for the light fermions satisfying M 2 < Λ/12, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the hamiltonian is discrete. We also prove a result of equipartition of the energy.

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