The linkage of renewable, non-renewable energy and carbon markets is increasing, and there is a complex network structure for the risk transmission among multiple markets. Based on the methods of network topology analysis and DY spillover index, this paper analyzes the time-varying spillover effect and network structure of risk transmission among renewable, non-renewable energy and carbon markets. The results show that: according to the static spillover index, there are significant spillover effects among renewable, non-renewable energy and carbon markets, and they are asymmetric. Moreover, the total spillover index further shows that the spillover effect between energy and carbon markets is time-varying, especially during the extreme events. Specifically, the net spillover index shows that the spillover effects among renewable, non-renewable energy and carbon markets are bidirectional, asymmetric and time-varying. Additionally, under the influence of various extreme events, the spillover effect and network structure of risk transmission among renewable, non-renewable energy and carbon markets are heterogeneous. Compared with the shale oil revolution and the Sino-US trade dispute, the influence of COVID-19 is more significant and complex, and it is long-term and comprehensive. Finally, some policy implications for preventing risk transmission and optimizing the energy structure to promote emission reduction are put forward.