The magnitude and nature of the COVID-19 pandemic prevents public health policies from relying on coercive enforcement. Practicing social distancing, wearing masks and staying at home are voluntary and conditional on the behavior of others. We present the results of a large-scale survey experiment in nine countries with representative samples of the population. We find that both empirical expectations (what others do) and normative expectations (what others approve of) play a significant role in compliance, beyond the effect of any other individual or group characteristic. In our survey experiment, when empirical and normative expectations of individuals are high, compliance goes up by 55% (relative to the low expectations condition). Similar results are obtained when we look at self-reported compliance among those with high expectations. Our results are driven by an asymmetric interaction with individuals’ trust in government and science. Holding both expectations high, the effect of trusting science is substantial and significant in our vignette experiment (22% increase in compliance), and even larger in self-reported compliance (76% and 127% increase before and after the lockdown). By contrast, trusting the government only generates modest effects. At the macro level, the country-level trust in science, and not in government, becomes a strong predictor of compliance.