All-electron ab initio Hartree-Fock (HF), valence configuration interaction (CI), and multiconfiguration self-consistent-field (CASSCF) calculations have been applied to investigate the electronic states of the CrC molecule. The molecule is predicted as having four low-lying electronic states, 3-SIGMA-, 5-SIGMA-, 7-SIGMA-, and 9-SIGMA-, separated by an energy gap of 0.55 eV from the next higher-lying state, 1-SIGMA-, which is followed by the states 5-PI and 7-PI. The four lowest-lying electronic states are due to the coupling of the angular momenta of the S-6(g) Cr+ ion with those of the S-4(u) C- anion. The chemical bond in the 3-SIGMA- ground state can be viewed as a quadruple bond composed of two sigma and two pi-bonds. One sigma-bond is due to the formation of a molecular orbital that is doubly occupied. The remaining bonds, i.e., one sigma and two pi-bonds, arise from valence-bond couplings. The pi-bonds originate from the valence-bond couplings of the electrons in the C 2p-pi orbitals with those in the Cr 3d-pi orbitals. The sigma-bond originates from the valence-bond coupling of the C 2p-sigma electron with an electron in the Cr 4s,4p hybrid that is polarized away from the C atom.