This study investigates the dynamic connectedness between stock indices and the effect of economic policy uncertainty (EPU) in eight countries where COVID-19 was most widespread (China, Italy, France, Germany, Spain, Russia, the US, and the UK) by implementing the time-varying VAR (TVP-VAR) model for daily data over the period spanning from 01/01/2015 to 05/18/2020. Results showed that stock markets were highly connected during the entire period, but the dynamic spillovers reached unprecedented heights during the COVID-19 pandemic in the first quarter of 2020. Moreover, we found that the European stock markets (except Italy) transmitted more spillovers to all other stock markets than they received, primarily during the COVID-19 outbreak. Further analysis using a nonlinear framework showed that the dynamic connectedness was more pronounced for negative than for positive returns. Also, findings showed that the direction of the EPU effect on net connectedness changed during the pandemic onset, indicating that information spillovers from a given market may signal either good or bad news for other markets, depending on the prevailing economic situation. These results have important implications for individual investors, portfolio managers, policymakers, investment banks, and central banks.