The impacts of COVID-19 have spread rapidly to global financial markets. In this context, combining the spillover index method introduced by Diebold and Yilmaz (2012) and the complex network analysis framework, we examined the volatility connectedness and the topological structure among the top ten cryptocurrencies before and during the COVID-19 crisis. The results revealed that the total volatility connectedness of the cryptocurrency market markedly increased following the outbreak of COVID-19; statically, Bitcoin, Ethereum, Cardano, and Bitcoin Cash were the net transmitters before COVID-19, while Bitcoin, Ethereum, Ripple, Litecoin, Cardano, and Stellar became the major net transmitters in the market after COVID-19. Dynamically, the dynamic performance of different cryptocurrencies during the COVID-19 pandemic was heterogeneous, and the possible driving factors are diverse. Moreover, from network analysis, we further found that the COVID-19 crisis has significantly changed the topological structure of the cryptocurrency market. Our findings may help understand the typical dynamics in the cryptocurrency market and provide significant implications for portfolio managers, investors, and government agencies in times of highly stressful events like the COVID-19 crisis.