Although the SARS-CoV-19 virus spread rapidly around in world in early 2020, disease epidemics in different places evolved differently as the year progressed - and the state of the COVID-19 pandemic now varies significantly across different countries and territories. We have created a taxonomy of possible categories of disease dynamics, and used the evolution of reported COVID-19 cases relative to changes in disease control measures, together with total reported cases and deaths, to allocate most countries and territories among the possible categories. As of 31 January 2021, we find that the disease was (1) kept out or suppressed quickly through quarantines and testing & tracing in 39 countries with 29 million people, (2) suppressed on one or more occasions through control measures in 74 countries with 2.49 billion people, (3) spread slowly but not suppressed, with cases still increasing or just past a peak, in 31 countries with 1.45 billion people, (4) spread through the population, but slowed a result of control measures, leading to a "flattened curve" and fewer infections than if the epidemic were unmitigated, in 32 countries with 2.24 billion people, and (5) spread through the population with some but limited mitigation in 5 countries with 168 million people. In addition, several countries have experienced increases in cases after disease appeared to have finished spreading due to declining numbers of susceptible people. For some of these countries - for example Kenya, Pakistan and Afghanistan - the resurgences can be explained by the relaxation of control measures (and may have been enhanced by disease spread in population segments that experienced lower infection levels during the first waves). For other countries, the resurgences point to the effects of new virus variants with higher-transmissibility or immunity resistance - including most countries in Southern Africa (where the B.1.351 variant has been identified) and several countries in West Africa (maybe due to the B.1.351 or a different variant). These findings are consistent with mounting evidence of high infection rates in several low- and middle-income countries, both from seroprevalence studies and estimates of actual deaths from COVID-19 combined with estimates of expected mortality rates. We estimate that 1.3-3.0 billion people, or 17-39% of the global population, have been infected by SARS-CoV-2 to date, and that at least 5 million people have died from COVID-19 - much higher than reported cases and deaths. Disease control policies and vaccination strategies should be designed based on the state of the COVID-19 epidemic in the population - and consequently may need to be different in different countries.