Complex networks can effectively describe interactions within real-world complex systems. In researches of epidemic spreading, scientists constructed various physical contact networks between individuals on the microscopic scale and the metapopulation networks on the macroscopic scale. These different types of network structures significantly impact the propagation dynamics of epidemic in human society. For instance, population flows in global airline networks influence the speed and arrival time of epidemics across large-scale space. In this paper we review the epidemic spreading models on various network structures, including fully mixed networks, three types of lower-order networks, three types of higher-order networks, metapopulation networks, and multiple strains competitive epidemic spreading models. We also provide an overview of the application of complex network theory in the COVID-19 pandemic, covering topics of prediction, prevention, and control of the epidemic. Finally, we discuss the strengths and limitations of these models and propose perspectives for future research.