People are concerned about false news online and media coverage of the problem is on the rise. Yet little is known about how these concerns have changed over time, and how they vary across countries and individuals. A pre-registered analysis of longitudinal data from 2018 to 2023 in 46 countries (N = 509,797) shows that concerns about false news have remained stable in most countries. Women, more educated, older, and right-wing respondents were more concerned about false news, together with respondents using social media for news and engaging with news more often (e.g., talking about news). Countries with higher levels of concern about false news had lower GDP per capita, lower press freedom, and higher levels of corruption. In these countries, respondents were more concerned about false news on messaging apps and false news coming from their government rather than from foreign governments. These findings shed light on the correlates of concerns about false news and could help design better targeted interventions against false news.