The electronic structure, elastic properties, Debye temperature and thermal conductivity of MgB2are investigated by using the first-principles density function theory within the generalized gradient approximation (GGA). The calculated elastic constants indicate that the MgB2is mechanically stable. The shear modulus, Young's modulus, Poisson's ratio, σ, the ratio B/G and universal anisotropy index are also calculated. Finally, the averaged sound velocity, longitudinal sound velocity, transverse sound velocity, Debye temperature and thermal conductivity are obtained.