Exchange anisotropy has been observed and investigated in single-crystalline CuO nanowires grown by thermal oxidation of Cu. The exchange bias field decreases by increasing temperature and can be tuned by the strength of the cooling field. A training effect has also been observed. The obtained results can be understood in terms of a phenomenological core-shell model, where the core of the CuO nanowire shows antiferromagnetic behavior and the surrounding shell behaves as a spin glass-like system due to uncompensated surface spins.