Theoretical predictions of the magnetic anisotropy of antiferromagnetic materials are demanding due to a lack of experimental techniques which are capable of a direct measurement of this quantity. At the same time it is highly significant due to the use of antiferromagnetic components in magneto-resistive sensor devices where the stability of the antiferromagnet is of upmost relevance. We perform an ab-initio study of the ordered phases of IrMn and IrMn3, the most widely used industrial antiferromagnets. Calculating the form and the strength of the magnetic anisotropy allows the construction of an effective spin model, which is tested against experimental measurements regarding the magnetic ground state and the Néel temperature. Our most important result is the extremely strong second order anisotropy for IrMn3 appearing in its frustrated triangular magnetic ground state, a surprising fact since the ordered L12 phase has a cubic symmetry. We explain this large anisotropy by the fact that cubic symmetry is locally broken for each of the three Mn sub-lattices. While the magnetic anisotropy (MA) of ferromagnets is a well investigated quantity, both experimentally as well as theoretically, it is much less understood in case of antiferromagnets. This lack of knowledge is on the one hand due to a lack of experimental techniques which are capable of a direct measurement of this quantity. On the other hand, theoretical first principles calculations of magnetic anisotropy effects are quite challenging as they require the use of fully relativistic spin density functional theory.Interest in the MA of antiferromagnets comes from the fact that these compounds are important components of GMR sensors used, e.g., in hard disc read heads. Antiferromagnetic materials are employed in these devices to form antiferromagnet/ferromagnet bilayers exhibiting exchange bias 1 , a shift of the hysteresis loop of the ferromagnet, providing a pinned layer which fixes the magnetization of the reference layer of a GMR sensor. The stability of the antiferromagnet is most crucial for the stability of exchange bias and hence the functioning of the device 2,3 . Industrially the antiferromagnet IrMn is widely used because of the large exchange bias and thermal stability that can be obtained with this material.From experimental investigations of the exchange bias effect it is concluded that IrMn must have a rather large MA. Recent estimates of the MA of IrMn concerned the mean blocking temperature T B , the temperature at which the exchange bias shift changes sign upon thermal activation. From T B the intrinsic MA can be inferred if the particle size distribution is known; such a procedure has recently been reported and the room temperature MA energy of IrMn was estimated at 5.5×10 6 erg/cc 4 and even 2.8×10 7 erg/cc 5 depending on the seed layer and, consequently, the texture of the IrMn.In this letter, we address several features of the MA of IrMn alloys starting from first principles. In terms of simple symmetry considerations we predict the form...