In a recent work Liu and Cohen (Phys. Rev. B., 41 (1990) 10 727-10 734) reported the theoretical prediction of a C-N phase whose hardness was competing with that of diamond. This work provoked extensive experimental search for the theoretical material with a composition β-C3N4. There was a report (Niu, Lu and Lieber, Science, 261 (1993) 334–337) on the growth β-C3N4 films. However, this has not been reproduced by others workers. Recently, Guo and Goddard (Chem. Phys. Lett., 237 (1995) 72) reported that the synthesized material, in most cases, corresponds to α-C3N4 and not to β-C3N4. We review the evidence from a number of papers and conclude that there is not enough evidence that C3N4 can be produced as large crystals. It is clear, however, that the α-C3N4 is more stable than the β-C3N4 phase. We discuss the recent report (Sarikaya, Qian, Martin-Gil, Martin-Gil and José-Yacamán, to be published) in which a new phase with the composition α-CxNy with x4 and y4 is found. We discuss the applications and future trends on these materials.