We estimate the effect of class size on student performance in 18 countries, combining school fixed effects and instrumental variables to identify random class-size variation between two adjacent grades within individual schools. Conventional estimates of class-size effects are shown to be severely biased by the non-random placement of students between and within schools. Smaller classes exhibit beneficial effects only in countries with relatively low teacher salaries. While we find sizable beneficial effects of smaller classes in Greece and Iceland, the possibility of even small effects is rejected in Japan and Singapore. In 11 countries, we rule out large class-size effects.