We present a detailed study of the structural and elastic properties of the half-Heusler compounds IrMnZ (Z = Al, Sn and Sb) using the full potential linearized augmented plane wave method within the generalized gradient approximation (GGA). The elastic constants are calculated for the energetically favourable magnetic structure. We have derived the bulk and the shear moduli, Young's and Poisson's ratio for poly-crystalline IrMnZ. The elastic modulus of IrMnAl is predicted to be the highest with a value of 152.48 GPa. Our results indicate a profound elastic anisotropy of the studied compounds. Finally, we have estimated the Debye temperatures for the IrMnZ compounds from the average sound velocity. (C) 2008 Elsevier B.V. All rights reserved.