As the COVID-19 pandemic has had a profound impact on public health and global economies in 2020; it is crucial to understand how it developed and spread in time and space. This paper contributes to the growing literature by considering the dynamics of country-wise growth rates of infection numbers. Low-order serial correlation of growth rates is predominantly negative with cycles of two to four days for most countries. The results of fitted spatial autoregressive models suggest that there is high degree of spillover between countries. Forecast variances of many countries, in particular those with a high absolute number of infections, can to a large extent be explained by structural innovations of other countries. A better understanding of the serial and spatial dynamics of the spread of the pandemic may contribute to an improved containment and risk management.