Carbon nitride in the form of C3N4 was speculated nearly 22 years ago. It has various structural forms ranging from layered graphitic to superhard structures. Using first principles calculations, we investigate the structural and elastic properties of these graphitic and superhard structures. The independent elastic constants of cubic, hexagonal, rhombohedral, buckled as well as orthorhombic phases of C3N4 are reported in this work. We also examine the bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio. From the work it is evident that although the compressibility of some of the superhard phases may surpass diamond, the shear modulus indicates that C3N4 is not harder than diamond, as had been speculated earlier.