In this paper, we investigate the dynamic connectedness between the stock indices in the eight most endemic countries by the COVID-19 (China, Italy, France, Germany, Spain, Russia, U.S., and the U.K) as well as the effect of the economic policy uncertainty (EPU) by implementing the TVP-VAR model for daily data over the period spanning from 01/01/2015 to 05/18/2020. Results show that stock markets are highly connected during the entire period, while the dynamic spillovers reached unprecedented heights during the COVID-19 pandemic in the first quarter of the year 2020. Moreover, we find that the European stock markets transmit more spillovers to all other stock markets more than they receive (except for Italy), mainly during the outbreak of the COVID-19 pandemic. Also, findings show that the sign of the effect of the EPU on the net connectedness changes during the onset of the pandemic, showing that information spillovers from a given market may be seen as good or bad news for other markets, given the economic situation prevailing. From the results of this study, important implications can be provided for individual investors, portfolio managers, policymakers, investment banks, and central banks.