The spreading andpropagation of viruses, rumors and opinions etc. on differenttypes of real-life networks affect significantly human activitiesin social and economic environments, and thus the study ondynamical behaviors of various spreading phenomena in complexnetworks has become one of the main subjects in the field ofcomplex network theory. This paper briefly surveys the latestprogresses of the study on spreading dynamics on complex dynamicalnetworks in the aspects of adaptive avoiding, response delay,discrete sampling and growing of open-systems, in an attempt toassist the investigations on dynamical processes of epidemicspreading behaviors on complex networks.