This paper investigates the impact of the COVID-19 pandemic on the volatility spillover and dependence structure among the major developed and emerging stock markets. The TVP-VAR connectedness decomposition approach and R-vine copula are implemented in this research. The results of the TVP-VAR connectedness decomposition approach reveal that the volatility spillover among the major developed and emerging stock markets has been significantly strengthened by the outbreak of the COVID-19 pandemic, although it has gradually faded over time. In addition, during the pandemic, the UK, German, French and Canadian stock markets are the spillover transmitters, while the Japanese, Chinese Hong Kong, Chinese and Indian stock markets are the receivers. It is also found that the US and Brazilian stock markets have undergone role shifts after the outbreak of the COVID-19 pandemic. The results of the R-vine copula model indicate that during the pandemic, the Canadian, French, and Chinese Hong Kong stock markets are the most important financial centre in the American, European, and Asian stock markets, respectively. Furthermore, the effect of the extreme risk contagion has been strengthened by the pandemic, particularly the downside risk contagion.