In this paper, the density functional theory (DFT) within the generalized gradient approximation (GGA) was used. The single crystal elastic constants for the intermetallic FeTi and its hydrides FeTiH and FeTiH2 are successfully obtained from the stress-strain relationship calculations and the strain energy-strain curves calculations, respectively. The shear modulus, Young's modulus, Poisson's ratio and shear anisotropic factors are also calculated. The bulk moduli derived from the elastic constants calculations of the cubic FeTi, orthorhombic P222(1) FeTiH and Cmmm FeTiH2 are calculated. For cubic FeTi compound, the bulk modulus is in a good agreement with both theoretical results and experimental data available in the literature. More importantly, it is found that, the insertion of hydrogen into the FeTi crystal structure causes an increase in the bulk modulus. From the analysis of shear-to-bulk modulus ratio, it is found that FeTi compound and its hydrides are ductile and that this ductibility, changes with changing the concentration of hydrogen. Copyright (C) 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.