It is unknown how widely the COVID-19 pandemic has spread: numbers of confirmed cases probably fall behind the total numbers of COVID-19 infections in many countries. However, it is critically important to know how many people are infected with COVID-19, how these infection numbers have developed over time, and what measures have been effective to prevent them from increasing further. We estimate the total numbers of COVID-19 infections with the demographic scaling model, which is (i) broadly applicable in contexts with both rich and poor data and (ii) accounts for cross-country differences in age structure, preconditions, and medical services. For our sample of ten countries with most COVID-19 deaths as of September-21-2020-located on 4 continents-we find, e.g., that the pandemic is far more widespread than officially confirmed and that infection numbers increase most strongly in India and take up speed again in Spain, France, and the U.K..