The thermodynamic stability, electronic structures and anisotropic elastic properties of Cr-B binary compounds are extensively investigated by means of the first principle calculations using the CASTEP code.Formation enthalpy is calculated and used to estimate the stability of the CrxBy compounds.The elastic constants are obtained by the stress-strain method and the Voigt-Reuss-Hill approximation is applied to evaluate the polycrystalline elastic modulus.Meanwhile, all the compounds show different mechanical anisotropy because of the calculated anisotropic index and different shape of surface contour for bulk modulus and Youngs modulus.CrB4 exhibits the largest value of hardness, Youngs modulus and shear modulus among all the CrxBy compounds.The total density of states and partial density of states are analyzed in physical properties of CrxBy compounds.Moreover, the Debye temperature, anisotropy of acoustic velocities and heat capacity at lower temperature are evaluated and discussed by semi-empirical methods.