Abstract The paper is concerned with the existence and global asymptotic stability of periodic solutions for a class of delayed complex-valued Cohen–Grossberg neural networks. Without using the method of the a priori estimate of periodic solutions, by combining Mawhin’s continuation theorem of coincidence degree theory with LMI method and using inequality techniques, a novel LMI-based sufficient condition on the existence of periodic solutions is established for the complex-valued Cohen–Grossberg neural networks. Then by using inequality techniques, a novel sufficient condition on the global asymptotic stability of periodic solutions for the above complex-valued neural networks is established. Our results and method are new and complementary to the existing papers on the study of periodic solutions of neural networks.