First-principles pseudopotential calculations of the lattice constants and of the single-crystal elastic constants for TixTa1-xN (0 <= x <= 1) alloys with BI-rocksalt structure were first carried out. These calculations were performed using density functional perturbation theory (DFPT) within the virtual crystal approximation (VCA) for the disordered alloys and the supercell method (SC) for the ordered alloys, with the ABINIT program. For disordered structures, partial comparisons of the lattice constants and of the bulk modulus are provided with calculations that used the coherent potential approximation (CPA) with the exact muffin-tin orbitals (EMTO). For the exchange-correlation potential we used the generalized gradient methods (GGA). The calculated equilibrium lattice parameters by VCA are in good agreement with the stress-free lattice parameters a(0) and exhibit a positive deviation from Vegard's rule corresponding to a positive bowing parameter while the calculated single-crystal stiffness c(12) and c(44), gradually increase when c(11) decreases from TaN to TiN. In a second stage, we have estimated by homogenization methods the averaged stiffnesses , the Young modulus and Poisson ratio of polycrystalline TixTa1-xN (0 <= x <= 1) alloys considering a random orientation of crystallites. Finally, comparisons are made with the experimental effective out-of-plane shear elastic modulus C-44 and the effective out-of-plane longitudinal elastic constant C-33 measured by Brillouin light scattering and picosecond ultrasonics, respectively, on thin films elaborated by magnetron sputtering. (C) 2012 Elsevier B.V. All rights reserved.