The neutron-rich cobalt isotopes up to A = 67 have been studied through multinucleon transfer reactions by bombarding a 238 U target with a 460-MeV 70 Zn beam. Unambiguous identification of prompt γ rays belonging to each nucleus has been achieved using coincidence relationships with the ions detected in a high-acceptance magnetic spectrometer. The new data are discussed in terms of the systematics of the cobalt isotopes and interpreted with large-scale shell-model calculations in the fpgd model space. In particular, very different shapes can be described in 67 Co, at the edge of the island of inversion at N = 40, where a low-lying highly deformed band coexists with a spherical structure.