We predict three novel phases of the carbon nitride (CN) bilayer, denoted α-C2N2, β-C2N2 and γ-C4N4, respectively. All of them consist of two CN sheets connected by C-C covalent bonds. The phonon dispersions reveal that all these phases are dynamically stable, because no imaginary frequency is present. The transition pathway between α-C2N2 and β-C2N2 is investigated, which involves bond-breaking and bond-reforming between C and N. This conversion is difficult, since the activation energy barrier is 1.90 eV per unit cell, high enough to prevent the transformation at room temperature. Electronic structure calculations show that all three phases are semiconductors with indirect band gaps of 3.76/5.22 eV, 4.23/5.75 eV and 2.06/3.53 eV, respectively, by PBE/HSE calculation. The β-C2N2 has the widest band gap among the three phases. All three bilayers can become metallic under tensile strain, and the indirect gap of γ-C4N4 can turn into a direct one. γ-C4N4 can become an anisotropic Dirac semimetal under uniaxial tensile strain. Anisotropic Dirac cones with high Fermi velocity of the order of 105 m/s appear under 12% strain. Our results suggest that the three two-dimensional materials have potential applications in electronics, semiconductors, optics and spintronics.