Four high-pressure N-rich compounds (Pmn21-CeN7, Amm2-CeN9, P1̄-CeN10, and P1̄-II-CeN10) are proposed using first-principles calculations. Novel polymeric units (a heart shaped layered structure, chain-like N8 rings, and two new banded structures) in four cerium nitrides are reported for the first time in this study. The analyses of electronic structures and bonding properties show that the charge transfer between Ce and N atoms promotes the formation of the Ce-N ionic bond and N-N covalent bond, which play an important role in stabilizing the nitrogen skeleton. Four new phases possess high energy densities (3.24-3.86 kJ g-1), indicating that they are favorable high-energy density materials. Moreover, P1̄-CeN10 possesses ultra-incompressibility along the [1 0 0] direction. Finally, infrared and Raman spectra are analyzed to provide guidance for experimental synthesis.