The spreading and propagation of viruses, rumors, and opinions, etc. on different categories of complex networks in real life significantly affect human activities in social and economic societies. The study of dynamical behaviors of various spreading phenomena on complex networks is one of main subjects in the research field of complex network theory. To investigate the dynamical processes of epidemic spreading behaviors on complex networks with the factors of adaptive avoiding, response delay, discrete sampling, and growing of open-system, in this talk, I will survey the latest progresses on the study of spreading dynamics on complex dynamical networks.