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The possible ground states of the undoped and doped Kitaev–Heisenberg model on a triangular lattice are studied. For the undoped system, a combination of the numerical exact diagonalization calculation and the four-sublattice transformation analysis suggests one possible exotic phase and four magnetically ordered phases, including a collinear stripe pattern and a noncollinear spiral pattern in the global phase diagram. The exotic phase near the antiferromagnetic (AF) Kitaev point is further investigated using the Schwinger-fermion mean-field method, and we obtain an energetically favorable Z _2 chiral spin liquid with a Chern number ±2 as a promising candidate. At finite doping, we find that the AF Heisenberg coupling supports an s -wave or a ${{d}_{{{x}^{2}}-{{y}^{2}}}}+{\rm i}{{d}_{xy}}$ -wave superconductivity (SC), while the AF and the ferromagnetic Kitaev interactions favor a ${{d}_{{{x}^{2}}-{{y}^{2}}}}+{\rm i}{{d}_{xy}}$ -wave SC and a time-reversal invariant topological p -wave SC, respectively. Possible experimental realizations and related candidate materials are also discussed.