The orientation dependence of the Young's and shear moduli for Re2N and Re3N was examined in basal planes, pyramidal planes, and prismatic planes, and was strongly orientation-dependent on the prismatic and pyramidal planes. The elastic constants of the rhenium nitrides are characterized by the large values of the Young moduli and the large ratios of the shear modulus to bulk modulus, which is a signature of brittle materials. Elastic anisotropy, the nature of chemical bonding, and the electronic charge transfer between constituent atoms have also been explored to assess the origins of high elastic stiffness of these compounds. Their properties can be explained from the strength of bonding in rhenium nitrides ranked in the ascending order as: Re-Re (metallic), Re-N (ionic), and N-N (covalent).