In this paper, a class of complex-valued neural networks with time-varying delays is studied. By employing an extension of Mawhin's continuation theorem and an approximation technique, several sufficient conditions of the new results on the existence of homoclinic solutions and periodic solutions are established. Moreover, the asymptotic behavior of solutions via the Lyapunov function is also investigated. Finally, for the purpose of validity, an example is given to illustrate the effectiveness of main results.